Abstract group 679392.d: $C_{4044}.C_{168}$

• Label: 679392.d
• Order: \( 2^{5} \cdot 3^{2} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $356$
• Trans deg.: $16176$
• Rank: $2$

Group information

Description:	$C_{4044}.C_{168}$
Order:	\(679392\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 337 \)
Exponent:	\(113232\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 337 \)
Automorphism group:	$C_{2022}.C_{168}.C_2^4$, of order \(5435136\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_7$, $C_{337}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336	337	674	1011	1348	2022	4044
Elements	1	675	2024	676	7416	2022	2696	9440	6066	5392	16176	21568	8088	48528	43136	16176	64704	32352	129408	258816	336	336	672	672	672	1344	679392
Conjugacy classes	1	3	8	4	24	6	8	32	18	16	48	64	24	144	128	48	192	96	384	768	2	2	4	4	4	8	2040
Divisions	1	3	4	2	12	1	2	8	3	2	4	8	2	12	8	2	8	2	8	8	1	1	1	1	1	1	106
Autjugacy classes	1	2	3	3	6	6	4	9	12	4	18	12	18	36	12	24	54	24	72	72	1	1	1	1	1	1	398

Minimal presentations

Permutation degree:	$356$
Transitive degree:	$16176$
Rank:	$2$
Inequivalent generating pairs:	$18432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	168	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{4044}=1, a^{168}=b^{2022}, b^{a}=b^{2113} \rangle$
Permutation group:	Degree $356$ $\langle(1,2,4,18,82,262,88,84,43,47,184,298,115,25,114,286,100,228,104,290,167,94,239,222,263,285,209,264,299,318,311,258,78,257,314,138,131,155,90,19,85,30,81,120,243,281,292,153,96,20,93,45,7,42,154,119,196,332,266,304,166,204,216,97,229,203,334,273,282,226,178,329,323,202,55,40,137,294,211,170,291,118,89,272,252,92,174,241,313,296,278,91,208,233,63,12,62,230,127,301,234,248,289,102,21,99,265,105,128,77,16,76,37,87,182,46,112,24,111,165,227,212,192,51,191,194,52,181,161,122,219,173,305,220,207,283,164,41,163,310,317,183,240,66,13,64,110,23,50,8,49,187,177,312,321,149,83,237,65,235,260,80,17,79,236,280,333,197,198,199,232,336,287,144,60,11,59,215,146,35,143,171,147,320,325,180,130,31,5,27,26,117,38,6,34,136,151,319,303,160,39,159,284,256,261,306,246,221,126,145,135,33,134,133,32,132,254,259,244,223,74,186,48,185,95,29,57,10,56,129,103,267,142,73,250,327,168,158,300,123,200,54,9,53,195,206,108,277,295,316,141,172,109,116,270,86,189,274,245,253,269,140,67,36,150,322,330,242,217,315,276,335,205,113,98,218,255,247,307,293,337,231,190,125,70,14,68,193,72,124,152,275,308,179,121,61,28,107,22,106,157,188,238,297,214,58,213,331,326,302,279,210,271,328,224,139,201,169,268,162,225,148,309,156,75,251,324,249,71,69,176,101,288,175,44,15,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 248 & 0 \\ 0 & 326 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 199 & 0 \\ 0 & 105 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_3$ $\, \times\, $ $(D_{337}:C_{336})$
Semidirect product:	$(D_{337}:C_{336})$ $\,\rtimes\,$ $C_3$	$(D_{337}:C_{48})$ $\,\rtimes\,$ $C_{21}$	$(D_{337}:C_{48})$ $\,\rtimes\,$ $C_{21}$ (3)	$(C_3\times D_{337})$ $\,\rtimes\,$ $C_{336}$ (2)	all 30
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{4044}$ . $C_{168}$	$(D_{337}:C_{168})$ . $C_6$	$(D_{337}:C_{168})$ . $C_6$ (3)	$(D_{337}:C_{24})$ . $C_{42}$	all 68

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{6} \times C_{336} \simeq C_{2} \times C_{16} \times C_{3}^{2} \times C_{7}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 54768 subgroups in 336 conjugacy classes, 174 normal (66 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $C_{337}:C_{168}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_6\times C_{336}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{2022}:C_{168}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4044}$	$G/\operatorname{Fit} \simeq$ $C_{168}$
Radical:	$R \simeq$ $C_{4044}.C_{168}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2022}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{168}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
337-Sylow subgroup:	$P_{ 337 } \simeq$ $C_{337}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 679392: $C_{4044}.C_{168}$

Order 339696: $C_{1011}:C_{336}$ x 2, $C_{4044}.C_{84}$

Order 226464: $D_{337}:C_{336}$ x 3, $D_{337}:C_{336}$

Order 169848: $C_{1011}:C_{168}$ x 2, $C_{4044}:C_{42}$

Order 113232: $C_{337}:C_{336}$ x 6, $D_{337}:C_{168}$ x 3, $C_{1011}:C_{112}$ x 2, $D_{337}:C_{168}$

Order 97056: $C_{4044}.C_{24}$

Order 84924: $C_{1011}:C_{84}$, $C_{2022}:C_{42}$, $C_{1011}:C_{84}$

Order 75488: $D_{337}:C_{112}$

Order 56616: $C_{337}:C_{168}$ x 6, $C_{1348}:C_{42}$ x 3, $C_{1011}:C_{56}$ x 2, $C_{1348}:C_{42}$

Order 48528: $C_{1011}:C_{48}$ x 2, $C_{4044}.C_{12}$

Order 42462: $C_{1011}:C_{42}$ x 2, $C_{1011}:C_{42}$

Order 37744: $C_{337}:C_{112}$ x 2, $D_{337}:C_{56}$

Order 32352: $D_{337}:C_{48}$ x 3, $D_{337}:C_{48}$

Order 28308: $C_{1348}:C_{21}$ x 3, $C_{674}:C_{42}$ x 3, $C_{337}:C_{84}$ x 3, $C_{1011}:C_{28}$, $C_{1011}:C_{28}$, $C_{2022}:C_{14}$

Order 24264: $C_{1011}:C_{24}$ x 2, $C_{4044}:C_6$

Order 21231: $C_{1011}:C_{21}$

Order 18872: $C_{337}:C_{56}$ x 2, $C_{1348}:C_{14}$

Order 16176: $C_{337}:C_{48}$ x 6, $D_{337}:C_{24}$ x 3, $C_{1011}:C_{16}$ x 2, $D_{337}:C_{24}$

Order 14154: $C_{337}:C_{42}$ x 6, $C_{337}:C_{42}$ x 3, $C_{1011}:C_{14}$ x 2, $C_{1011}:C_{14}$

Order 12132: $C_{1011}:C_{12}$, $C_{1011}:C_{12}$, $C_{2022}:C_6$

Order 10784: $D_{337}:C_{16}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 8088: $C_{337}:C_{24}$ x 6, $C_{1348}:C_6$ x 3, $C_{1011}:C_8$ x 2, $C_{12}\times D_{337}$

Order 7077: $C_{337}:C_{21}$ x 3, $C_{1011}:C_7$

Order 6066: $C_{1011}:C_6$ x 2, $C_{1011}:C_6$

Order 5392: $C_{337}:C_{16}$ x 2, $D_{337}:C_8$

Order 4718: $C_{337}:C_{14}$ x 2, $C_{337}:C_{14}$

Order 4044: $C_{337}:C_{12}$ x 3, $C_{674}:C_6$ x 3, $C_{1348}:C_3$ x 3, $C_3\times D_{674}$, $C_{1011}:C_4$, $C_{4044}$

Order 3033: $C_{1011}:C_3$

Order 2696: $C_{337}:C_8$ x 2, $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{337}:C_6$ x 6, $C_{337}:C_6$ x 3, $C_3\times D_{337}$ x 2, $C_{2022}$

Order 2016: $C_6\times C_{336}$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 1011: $C_{337}:C_3$ x 3, $C_{1011}$

Order 1008: $C_3\times C_{336}$ x 2, $C_6\times C_{168}$

Order 674: $D_{337}$ x 2, $C_{674}$

Order 672: $C_2\times C_{336}$ x 4

Order 504: $C_3\times C_{168}$ x 2, $C_6\times C_{84}$

Order 337: $C_{337}$

Order 336: $C_{336}$ x 8, $C_2\times C_{168}$ x 4

Order 288: $C_6\times C_{48}$

Order 252: $C_3\times C_{84}$ x 2, $C_6\times C_{42}$

Order 224: $C_2\times C_{112}$

Order 168: $C_{168}$ x 8, $C_2\times C_{84}$ x 4

Order 144: $C_3\times C_{48}$ x 2, $C_6\times C_{24}$

Order 126: $C_3\times C_{42}$ x 3

Order 112: $C_{112}$ x 2, $C_2\times C_{56}$

Order 96: $C_2\times C_{48}$ x 4

Order 84: $C_{84}$ x 8, $C_2\times C_{42}$ x 4

Order 72: $C_3\times C_{24}$ x 2, $C_6\times C_{12}$

Order 63: $C_3\times C_{21}$

Order 56: $C_{56}$ x 2, $C_2\times C_{28}$

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 4

Order 42: $C_{42}$ x 12

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $C_2\times C_{16}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_{24}$ x 8, $C_2\times C_{12}$ x 4

Order 21: $C_{21}$ x 4

Order 18: $C_3\times C_6$ x 3

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 12

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 679392: $C_{4044}.C_{168}$

Order 339696: $C_{4044}.C_{84}$, $C_{1011}:C_{336}$

Order 226464: $D_{337}:C_{336}$, $D_{337}:C_{336}$

Order 169848: $C_{1011}:C_{168}$ x 2, $C_{4044}:C_{42}$

Order 113232: $C_{1011}:C_{112}$, $D_{337}:C_{168}$, $D_{337}:C_{168}$, $C_{337}:C_{336}$

Order 97056: $C_{4044}.C_{24}$

Order 84924: $C_{1011}:C_{84}$, $C_{2022}:C_{42}$, $C_{1011}:C_{84}$

Order 75488: $D_{337}:C_{112}$

Order 56616: $C_{1011}:C_{56}$ x 2, $C_{337}:C_{168}$ x 2, $C_{1348}:C_{42}$, $C_{1348}:C_{42}$

Order 48528: $C_{1011}:C_{48}$, $C_{4044}.C_{12}$

Order 42462: $C_{1011}:C_{42}$, $C_{1011}:C_{42}$

Order 37744: $D_{337}:C_{56}$, $C_{337}:C_{112}$

Order 32352: $D_{337}:C_{48}$, $D_{337}:C_{48}$

Order 28308: $C_{1011}:C_{28}$, $C_{1011}:C_{28}$, $C_{1348}:C_{21}$, $C_{2022}:C_{14}$, $C_{674}:C_{42}$, $C_{337}:C_{84}$

Order 24264: $C_{1011}:C_{24}$ x 2, $C_{4044}:C_6$

Order 21231: $C_{1011}:C_{21}$

Order 18872: $C_{337}:C_{56}$ x 2, $C_{1348}:C_{14}$

Order 16176: $D_{337}:C_{24}$, $C_{337}:C_{48}$, $C_{1011}:C_{16}$, $D_{337}:C_{24}$

Order 14154: $C_{337}:C_{42}$, $C_{1011}:C_{14}$, $C_{337}:C_{42}$, $C_{1011}:C_{14}$

Order 12132: $C_{1011}:C_{12}$, $C_{1011}:C_{12}$, $C_{2022}:C_6$

Order 10784: $D_{337}:C_{16}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 8088: $C_{1011}:C_8$ x 2, $C_{337}:C_{24}$ x 2, $C_{12}\times D_{337}$, $C_{1348}:C_6$

Order 7077: $C_{1011}:C_7$, $C_{337}:C_{21}$

Order 6066: $C_{1011}:C_6$, $C_{1011}:C_6$

Order 5392: $C_{337}:C_{16}$, $D_{337}:C_8$

Order 4718: $C_{337}:C_{14}$, $C_{337}:C_{14}$

Order 4044: $C_3\times D_{674}$, $C_{1011}:C_4$, $C_{337}:C_{12}$, $C_{674}:C_6$, $C_{1348}:C_3$, $C_{4044}$

Order 3033: $C_{1011}:C_3$

Order 2696: $C_{337}:C_8$ x 2, $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{2022}$, $C_3\times D_{337}$, $C_{337}:C_6$, $C_{337}:C_6$

Order 2016: $C_6\times C_{336}$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 1011: $C_{1011}$, $C_{337}:C_3$

Order 1008: $C_6\times C_{168}$, $C_3\times C_{336}$

Order 674: $C_{674}$, $D_{337}$

Order 672: $C_2\times C_{336}$ x 2

Order 504: $C_3\times C_{168}$ x 2, $C_6\times C_{84}$

Order 337: $C_{337}$

Order 336: $C_{336}$ x 2, $C_2\times C_{168}$ x 2

Order 288: $C_6\times C_{48}$

Order 252: $C_3\times C_{84}$ x 2, $C_6\times C_{42}$

Order 224: $C_2\times C_{112}$

Order 168: $C_{168}$ x 4, $C_2\times C_{84}$ x 2

Order 144: $C_3\times C_{48}$, $C_6\times C_{24}$

Order 126: $C_3\times C_{42}$ x 2

Order 112: $C_2\times C_{56}$, $C_{112}$

Order 96: $C_2\times C_{48}$ x 2

Order 84: $C_{84}$ x 4, $C_2\times C_{42}$ x 2

Order 72: $C_3\times C_{24}$ x 2, $C_6\times C_{12}$

Order 63: $C_3\times C_{21}$

Order 56: $C_{56}$ x 2, $C_2\times C_{28}$

Order 48: $C_2\times C_{24}$ x 2, $C_{48}$ x 2

Order 42: $C_{42}$ x 4

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $C_2\times C_{16}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 2

Order 21: $C_{21}$ x 2

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2\times C_8$, $C_{16}$

Order 14: $C_{14}$ x 2

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 4

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 679392: $C_{4044}.C_{168}$ ($C_1$)

Order 339696: $C_{1011}:C_{336}$ ($C_2$) x 2, $C_{4044}.C_{84}$ ($C_2$)

Order 226464: $D_{337}:C_{336}$ ($C_3$) x 3, $D_{337}:C_{336}$ ($C_3$)

Order 169848: $C_{4044}:C_{42}$ ($C_4$), $C_{1011}:C_{168}$ ($C_2^2$), $C_{1011}:C_{168}$ ($C_4$)

Order 113232: $C_{337}:C_{336}$ ($C_6$) x 6, $D_{337}:C_{168}$ ($C_6$) x 3, $C_{1011}:C_{112}$ ($C_6$) x 2, $D_{337}:C_{168}$ ($C_6$)

Order 97056: $C_{4044}.C_{24}$ ($C_7$)

Order 84924: $C_{1011}:C_{84}$ ($C_8$), $C_{2022}:C_{42}$ ($C_8$), $C_{1011}:C_{84}$ ($C_2\times C_4$)

Order 75488: $D_{337}:C_{112}$ ($C_3^2$)

Order 56616: $C_{1348}:C_{42}$ ($C_{12}$) x 3, $C_{337}:C_{168}$ ($C_2\times C_6$) x 3, $C_{337}:C_{168}$ ($C_{12}$) x 3, $C_{1348}:C_{42}$ ($C_{12}$), $C_{1011}:C_{56}$ ($C_2\times C_6$), $C_{1011}:C_{56}$ ($C_{12}$)

Order 48528: $C_{1011}:C_{48}$ ($C_{14}$) x 2, $C_{4044}.C_{12}$ ($C_{14}$)

Order 42462: $C_{1011}:C_{42}$ ($C_{16}$) x 2, $C_{1011}:C_{42}$ ($C_2\times C_8$)

Order 37744: $C_{337}:C_{112}$ ($C_3\times C_6$) x 2, $D_{337}:C_{56}$ ($C_3\times C_6$)

Order 32352: $D_{337}:C_{48}$ ($C_{21}$) x 3, $D_{337}:C_{48}$ ($C_{21}$)

Order 28308: $C_{1348}:C_{21}$ ($C_{24}$) x 3, $C_{674}:C_{42}$ ($C_{24}$) x 3, $C_{337}:C_{84}$ ($C_2\times C_{12}$) x 3, $C_{1011}:C_{28}$ ($C_{24}$), $C_{1011}:C_{28}$ ($C_2\times C_{12}$), $C_{2022}:C_{14}$ ($C_{24}$)

Order 24264: $C_{4044}:C_6$ ($C_{28}$), $C_{1011}:C_{24}$ ($C_2\times C_{14}$), $C_{1011}:C_{24}$ ($C_{28}$)

Order 21231: $C_{1011}:C_{21}$ ($C_2\times C_{16}$)

Order 18872: $C_{1348}:C_{14}$ ($C_3\times C_{12}$), $C_{337}:C_{56}$ ($C_3\times C_{12}$), $C_{337}:C_{56}$ ($C_6^2$)

Order 16176: $C_{337}:C_{48}$ ($C_{42}$) x 6, $D_{337}:C_{24}$ ($C_{42}$) x 3, $C_{1011}:C_{16}$ ($C_{42}$) x 2, $D_{337}:C_{24}$ ($C_{42}$)

Order 14154: $C_{337}:C_{42}$ ($C_{48}$) x 6, $C_{337}:C_{42}$ ($C_2\times C_{24}$) x 3, $C_{1011}:C_{14}$ ($C_{48}$) x 2, $C_{1011}:C_{14}$ ($C_2\times C_{24}$)

Order 12132: $C_{1011}:C_{12}$ ($C_2\times C_{28}$), $C_{1011}:C_{12}$ ($C_{56}$), $C_{2022}:C_6$ ($C_{56}$)

Order 10784: $D_{337}:C_{16}$ ($C_3\times C_{21}$)

Order 9436: $C_{337}:C_{28}$ ($C_6\times C_{12}$), $C_{674}:C_{14}$ ($C_3\times C_{24}$), $C_{1348}:C_7$ ($C_3\times C_{24}$)

Order 8088: $C_{337}:C_{24}$ ($C_{84}$) x 3, $C_{337}:C_{24}$ ($C_2\times C_{42}$) x 3, $C_{1348}:C_6$ ($C_{84}$) x 3, $C_{1011}:C_8$ ($C_{84}$), $C_{1011}:C_8$ ($C_2\times C_{42}$), $C_{12}\times D_{337}$ ($C_{84}$)

Order 7077: $C_{337}:C_{21}$ ($C_2\times C_{48}$) x 3, $C_{1011}:C_7$ ($C_2\times C_{48}$)

Order 6066: $C_{1011}:C_6$ ($C_{112}$) x 2, $C_{1011}:C_6$ ($C_2\times C_{56}$)

Order 5392: $C_{337}:C_{16}$ ($C_3\times C_{42}$) x 2, $D_{337}:C_8$ ($C_3\times C_{42}$)

Order 4718: $C_{337}:C_{14}$ ($C_3\times C_{48}$) x 2, $C_{337}:C_{14}$ ($C_6\times C_{24}$)

Order 4044: $C_{337}:C_{12}$ ($C_2\times C_{84}$) x 3, $C_{674}:C_6$ ($C_{168}$) x 3, $C_{1348}:C_3$ ($C_{168}$) x 3, $C_3\times D_{674}$ ($C_{168}$), $C_{1011}:C_4$ ($C_2\times C_{84}$), $C_{4044}$ ($C_{168}$)

Order 3033: $C_{1011}:C_3$ ($C_2\times C_{112}$)

Order 2696: $C_4\times D_{337}$ ($C_3\times C_{84}$), $C_{337}:C_8$ ($C_6\times C_{42}$), $C_{337}:C_8$ ($C_3\times C_{84}$)

Order 2359: $C_{337}:C_7$ ($C_6\times C_{48}$)

Order 2022: $C_{337}:C_6$ ($C_{336}$) x 6, $C_{337}:C_6$ ($C_2\times C_{168}$) x 3, $C_3\times D_{337}$ ($C_{336}$) x 2, $C_{2022}$ ($C_2\times C_{168}$)

Order 1348: $D_{674}$ ($C_3\times C_{168}$), $C_{1348}$ ($C_3\times C_{168}$), $C_{337}:C_4$ ($C_6\times C_{84}$)

Order 1011: $C_{337}:C_3$ ($C_2\times C_{336}$) x 3, $C_{1011}$ ($C_2\times C_{336}$)

Order 674: $D_{337}$ ($C_3\times C_{336}$) x 2, $C_{674}$ ($C_6\times C_{168}$)

Order 337: $C_{337}$ ($C_6\times C_{336}$)

Order 12: $C_{12}$ ($C_{337}:C_{168}$)

Order 6: $C_6$ ($C_{674}:C_{168}$)

Order 4: $C_4$ ($C_{1011}:C_{168}$)

Order 3: $C_3$ ($D_{337}:C_{336}$)

Order 2: $C_2$ ($C_{2022}:C_{168}$)

Order 1: $C_1$ ($C_{4044}.C_{168}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 679392: $C_{4044}.C_{168}$ ($C_1$)

Order 339696: $C_{4044}.C_{84}$ ($C_2$), $C_{1011}:C_{336}$ ($C_2$)

Order 226464: $D_{337}:C_{336}$ ($C_3$), $D_{337}:C_{336}$ ($C_3$)

Order 169848: $C_{4044}:C_{42}$ ($C_4$), $C_{1011}:C_{168}$ ($C_2^2$), $C_{1011}:C_{168}$ ($C_4$)

Order 113232: $C_{1011}:C_{112}$ ($C_6$), $D_{337}:C_{168}$ ($C_6$), $D_{337}:C_{168}$ ($C_6$), $C_{337}:C_{336}$ ($C_6$)

Order 97056: $C_{4044}.C_{24}$ ($C_7$)

Order 84924: $C_{1011}:C_{84}$ ($C_8$), $C_{2022}:C_{42}$ ($C_8$), $C_{1011}:C_{84}$ ($C_2\times C_4$)

Order 75488: $D_{337}:C_{112}$ ($C_3^2$)

Order 56616: $C_{1348}:C_{42}$ ($C_{12}$), $C_{1011}:C_{56}$ ($C_2\times C_6$), $C_{1011}:C_{56}$ ($C_{12}$), $C_{1348}:C_{42}$ ($C_{12}$), $C_{337}:C_{168}$ ($C_2\times C_6$), $C_{337}:C_{168}$ ($C_{12}$)

Order 48528: $C_{1011}:C_{48}$ ($C_{14}$), $C_{4044}.C_{12}$ ($C_{14}$)

Order 42462: $C_{1011}:C_{42}$ ($C_{16}$), $C_{1011}:C_{42}$ ($C_2\times C_8$)

Order 37744: $D_{337}:C_{56}$ ($C_3\times C_6$), $C_{337}:C_{112}$ ($C_3\times C_6$)

Order 32352: $D_{337}:C_{48}$ ($C_{21}$), $D_{337}:C_{48}$ ($C_{21}$)

Order 28308: $C_{1011}:C_{28}$ ($C_{24}$), $C_{1011}:C_{28}$ ($C_2\times C_{12}$), $C_{1348}:C_{21}$ ($C_{24}$), $C_{2022}:C_{14}$ ($C_{24}$), $C_{674}:C_{42}$ ($C_{24}$), $C_{337}:C_{84}$ ($C_2\times C_{12}$)

Order 24264: $C_{4044}:C_6$ ($C_{28}$), $C_{1011}:C_{24}$ ($C_2\times C_{14}$), $C_{1011}:C_{24}$ ($C_{28}$)

Order 21231: $C_{1011}:C_{21}$ ($C_2\times C_{16}$)

Order 18872: $C_{1348}:C_{14}$ ($C_3\times C_{12}$), $C_{337}:C_{56}$ ($C_3\times C_{12}$), $C_{337}:C_{56}$ ($C_6^2$)

Order 16176: $D_{337}:C_{24}$ ($C_{42}$), $C_{337}:C_{48}$ ($C_{42}$), $C_{1011}:C_{16}$ ($C_{42}$), $D_{337}:C_{24}$ ($C_{42}$)

Order 14154: $C_{337}:C_{42}$ ($C_{48}$), $C_{1011}:C_{14}$ ($C_{48}$), $C_{337}:C_{42}$ ($C_2\times C_{24}$), $C_{1011}:C_{14}$ ($C_2\times C_{24}$)

Order 12132: $C_{1011}:C_{12}$ ($C_2\times C_{28}$), $C_{1011}:C_{12}$ ($C_{56}$), $C_{2022}:C_6$ ($C_{56}$)

Order 10784: $D_{337}:C_{16}$ ($C_3\times C_{21}$)

Order 9436: $C_{337}:C_{28}$ ($C_6\times C_{12}$), $C_{674}:C_{14}$ ($C_3\times C_{24}$), $C_{1348}:C_7$ ($C_3\times C_{24}$)

Order 8088: $C_{1011}:C_8$ ($C_{84}$), $C_{1011}:C_8$ ($C_2\times C_{42}$), $C_{12}\times D_{337}$ ($C_{84}$), $C_{337}:C_{24}$ ($C_{84}$), $C_{337}:C_{24}$ ($C_2\times C_{42}$), $C_{1348}:C_6$ ($C_{84}$)

Order 7077: $C_{1011}:C_7$ ($C_2\times C_{48}$), $C_{337}:C_{21}$ ($C_2\times C_{48}$)

Order 6066: $C_{1011}:C_6$ ($C_2\times C_{56}$), $C_{1011}:C_6$ ($C_{112}$)

Order 5392: $C_{337}:C_{16}$ ($C_3\times C_{42}$), $D_{337}:C_8$ ($C_3\times C_{42}$)

Order 4718: $C_{337}:C_{14}$ ($C_6\times C_{24}$), $C_{337}:C_{14}$ ($C_3\times C_{48}$)

Order 4044: $C_3\times D_{674}$ ($C_{168}$), $C_{1011}:C_4$ ($C_2\times C_{84}$), $C_{337}:C_{12}$ ($C_2\times C_{84}$), $C_{674}:C_6$ ($C_{168}$), $C_{1348}:C_3$ ($C_{168}$), $C_{4044}$ ($C_{168}$)

Order 3033: $C_{1011}:C_3$ ($C_2\times C_{112}$)

Order 2696: $C_4\times D_{337}$ ($C_3\times C_{84}$), $C_{337}:C_8$ ($C_6\times C_{42}$), $C_{337}:C_8$ ($C_3\times C_{84}$)

Order 2359: $C_{337}:C_7$ ($C_6\times C_{48}$)

Order 2022: $C_{2022}$ ($C_2\times C_{168}$), $C_3\times D_{337}$ ($C_{336}$), $C_{337}:C_6$ ($C_2\times C_{168}$), $C_{337}:C_6$ ($C_{336}$)

Order 1348: $D_{674}$ ($C_3\times C_{168}$), $C_{1348}$ ($C_3\times C_{168}$), $C_{337}:C_4$ ($C_6\times C_{84}$)

Order 1011: $C_{1011}$ ($C_2\times C_{336}$), $C_{337}:C_3$ ($C_2\times C_{336}$)

Order 674: $C_{674}$ ($C_6\times C_{168}$), $D_{337}$ ($C_3\times C_{336}$)

Order 337: $C_{337}$ ($C_6\times C_{336}$)

Order 12: $C_{12}$ ($C_{337}:C_{168}$)

Order 6: $C_6$ ($C_{674}:C_{168}$)

Order 4: $C_4$ ($C_{1011}:C_{168}$)

Order 3: $C_3$ ($D_{337}:C_{336}$)

Order 2: $C_2$ ($C_{2022}:C_{168}$)

Order 1: $C_1$ ($C_{4044}.C_{168}$)

Series

Derived series

$C_{4044}.C_{168}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{4044}.C_{168}$

$\rhd$

$C_{4044}.C_{84}$

$\rhd$

$C_{4044}:C_{42}$

$\rhd$

$C_{1011}:C_{84}$

$\rhd$

$C_{1011}:C_{28}$

$\rhd$

$C_{4044}$

$\rhd$

$C_{2022}$

$\rhd$

$C_{1011}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{4044}.C_{168}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $2040 \times 2040$ character table is not available for this group.

Rational character table

The $106 \times 106$ rational character table is not available for this group.