Abstract group 169848.a: $C_{1011}:C_{168}$

• Label: 169848.a
• Order: \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $348$
• Trans deg.: $8088$
• Rank: $2$

Group information

Description:	$C_{1011}:C_{168}$
Order:	\(169848\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 337 \)
Exponent:	\(56616\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 337 \)
Automorphism group:	$C_{1011}.C_{168}.C_2^3$, of order \(1358784\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 3, $C_3$ 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	21	24	28	42	56	84	168	337	674	1011	2022
Elements	1	1	2024	674	2024	2022	1348	5392	2022	16176	10784	4044	16176	8088	32352	64704	336	336	672	672	169848
Conjugacy classes	1	1	8	2	8	6	4	16	6	48	32	12	48	24	96	192	4	4	8	8	528
Divisions	1	1	4	1	4	1	1	4	1	4	4	1	4	1	4	4	1	1	1	1	44
Autjugacy classes	1	1	3	2	3	6	2	6	6	18	6	12	18	12	36	36	1	1	1	1	172

Dimension	1	2	4	6	8	12	24	48	84	336	672
Irr. complex chars.	504	0	0	0	0	0	0	0	24	0	0	528
Irr. rational chars.	2	9	5	2	4	9	5	4	0	2	2	44

Minimal presentations

Permutation degree:	$348$
Transitive degree:	$8088$
Rank:	$2$
Inequivalent generating pairs:	$9216$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	84	not computed	672
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{168}=b^{1011}=1, b^{a}=b^{82} \rangle$
Permutation group:	Degree $348$ $\langle(1,2,4,14,52,164,171,203,95,26,94,78,21,5,18,66,155,131,59,185,218,74,177,256,99,28,7,27,96,251,188,295,309,247,237,86,121,147,168,53,167,181,89,242,179,240,286,126,284,166,175,61,16,60,186,224,77,108,243,235,321,234,109,204,231,322,336,268,239,288,160,50,13,49,81,22,80,170,54,150,182,57,180,315,183,222,311,267,106,265,334,221,91,245,312,337,281,282,217,307,333,263,139,42,137,293,233,285,151,279,119,277,255,172,248,92,25,6,23,83,174,55,173,273,111,254,97,132,39,32,113,165,271,266,269,187,158,195,215,300,331,262,287,149,45,148,226,79,114,103,29,102,259,324,283,125,76,223,329,244,90,161,118,124,35,122,280,120,34,19,69,208,75,20,73,70,107,129,289,323,276,115,67,202,330,236,84,48,154,88,117,196,64,17,63,194,241,328,178,232,227,112,264,320,317,314,211,163,51,24,87,152,310,157,219,306,144,299,134,40,10,38,101,258,305,169,213,71,189,98,206,298,184,58,15,56,176,327,190,230,214,220,326,252,291,130,261,325,318,159,316,253,260,209,272,210,138,199,65,143,250,128,37,110,31,8,30,105,201,296,319,335,332,212,197,156,225,270,216,290,162,116,278,275,146,44,145,257,229,207,68,205,297,301,249,133,200,303,141,192,62,72,135,294,292,193,238,308,313,153,47,12,46,100,228,191,123,246,302,140,93,104,127,36,9,33,85,136,41,82,198,274,304,142,43,11,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 170 & 0 \\ 0 & 121 \end{array}\right), \left(\begin{array}{rr} 172 & 0 \\ 0 & 241 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_3$ $\, \times\, $ $(C_{337}:C_{168})$
Semidirect product:	$C_{1011}$ $\,\rtimes\,$ $C_{168}$	$(C_{1011}:C_{56})$ $\,\rtimes\,$ $C_3$	$(C_{1011}:C_{24})$ $\,\rtimes\,$ $C_7$	$(C_{1011}:C_{21})$ $\,\rtimes\,$ $C_8$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{2022}$ . $C_{84}$	$(C_{337}:C_{84})$ . $C_6$ (3)	$(C_{337}:C_6)$ . $C_{84}$ (3)	$C_{674}$ . $(C_3\times C_{84})$	all 18

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

Homology

Abelianization:	$C_{3} \times C_{168} \simeq C_{8} \times C_{3}^{2} \times C_{7}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 14880 subgroups in 96 conjugacy classes, 52 normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_{337}:C_{84}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_3\times C_{168}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{1011}:C_{84}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2022}$	$G/\operatorname{Fit} \simeq$ $C_{84}$
Radical:	$R \simeq$ $C_{1011}:C_{168}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2022}$	$G/\operatorname{soc} \simeq$ $C_{84}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
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

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 169848: $C_{1011}:C_{168}$

Order 84924: $C_{1011}:C_{84}$

Order 56616: $C_{337}:C_{168}$ x 3, $C_{1011}:C_{56}$

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

Order 28308: $C_{337}:C_{84}$ x 3, $C_{1011}:C_{28}$

Order 24264: $C_{1011}:C_{24}$

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

Order 18872: $C_{337}:C_{56}$

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

Order 12132: $C_{1011}:C_{12}$

Order 9436: $C_{337}:C_{28}$

Order 8088: $C_{337}:C_{24}$ x 3, $C_{1011}:C_8$

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

Order 6066: $C_{1011}:C_6$

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

Order 4044: $C_{337}:C_{12}$ x 3, $C_{1011}:C_4$

Order 3033: $C_{1011}:C_3$

Order 2696: $C_{337}:C_8$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{337}:C_6$ x 3, $C_{2022}$

Order 1348: $C_{337}:C_4$

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

Order 674: $C_{674}$

Order 504: $C_3\times C_{168}$

Order 337: $C_{337}$

Order 252: $C_3\times C_{84}$

Order 168: $C_{168}$ x 4

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

Order 84: $C_{84}$ x 4

Order 72: $C_3\times C_{24}$

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

Order 56: $C_{56}$

Order 42: $C_{42}$ x 4

Order 36: $C_3\times C_{12}$

Order 28: $C_{28}$

Order 24: $C_{24}$ x 4

Order 21: $C_{21}$ x 4

Order 18: $C_3\times C_6$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 169848: $C_{1011}:C_{168}$

Order 84924: $C_{1011}:C_{84}$

Order 56616: $C_{1011}:C_{56}$, $C_{337}:C_{168}$

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

Order 28308: $C_{1011}:C_{28}$, $C_{337}:C_{84}$

Order 24264: $C_{1011}:C_{24}$

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

Order 18872: $C_{337}:C_{56}$

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

Order 12132: $C_{1011}:C_{12}$

Order 9436: $C_{337}:C_{28}$

Order 8088: $C_{1011}:C_8$, $C_{337}:C_{24}$

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

Order 6066: $C_{1011}:C_6$

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

Order 4044: $C_{1011}:C_4$, $C_{337}:C_{12}$

Order 3033: $C_{1011}:C_3$

Order 2696: $C_{337}:C_8$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{2022}$, $C_{337}:C_6$

Order 1348: $C_{337}:C_4$

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

Order 674: $C_{674}$

Order 504: $C_3\times C_{168}$

Order 337: $C_{337}$

Order 252: $C_3\times C_{84}$

Order 168: $C_{168}$ x 2

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

Order 84: $C_{84}$ x 2

Order 72: $C_3\times C_{24}$

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

Order 56: $C_{56}$

Order 42: $C_{42}$ x 2

Order 36: $C_3\times C_{12}$

Order 28: $C_{28}$

Order 24: $C_{24}$ x 2

Order 21: $C_{21}$ x 2

Order 18: $C_3\times C_6$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 2

Order 4: $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 169848: $C_{1011}:C_{168}$ ($C_1$)

Order 84924: $C_{1011}:C_{84}$ ($C_2$)

Order 56616: $C_{337}:C_{168}$ ($C_3$) x 3, $C_{1011}:C_{56}$ ($C_3$)

Order 42462: $C_{1011}:C_{42}$ ($C_4$)

Order 28308: $C_{337}:C_{84}$ ($C_6$) x 3, $C_{1011}:C_{28}$ ($C_6$)

Order 24264: $C_{1011}:C_{24}$ ($C_7$)

Order 21231: $C_{1011}:C_{21}$ ($C_8$)

Order 18872: $C_{337}:C_{56}$ ($C_3^2$)

Order 14154: $C_{337}:C_{42}$ ($C_{12}$) x 3, $C_{1011}:C_{14}$ ($C_{12}$)

Order 12132: $C_{1011}:C_{12}$ ($C_{14}$)

Order 9436: $C_{337}:C_{28}$ ($C_3\times C_6$)

Order 8088: $C_{337}:C_{24}$ ($C_{21}$) x 3, $C_{1011}:C_8$ ($C_{21}$)

Order 7077: $C_{337}:C_{21}$ ($C_{24}$) x 3, $C_{1011}:C_7$ ($C_{24}$)

Order 6066: $C_{1011}:C_6$ ($C_{28}$)

Order 4718: $C_{337}:C_{14}$ ($C_3\times C_{12}$)

Order 4044: $C_{337}:C_{12}$ ($C_{42}$) x 3, $C_{1011}:C_4$ ($C_{42}$)

Order 3033: $C_{1011}:C_3$ ($C_{56}$)

Order 2696: $C_{337}:C_8$ ($C_3\times C_{21}$)

Order 2359: $C_{337}:C_7$ ($C_3\times C_{24}$)

Order 2022: $C_{337}:C_6$ ($C_{84}$) x 3, $C_{2022}$ ($C_{84}$)

Order 1348: $C_{337}:C_4$ ($C_3\times C_{42}$)

Order 1011: $C_{337}:C_3$ ($C_{168}$) x 3, $C_{1011}$ ($C_{168}$)

Order 674: $C_{674}$ ($C_3\times C_{84}$)

Order 337: $C_{337}$ ($C_3\times C_{168}$)

Order 6: $C_6$ ($C_{337}:C_{84}$)

Order 3: $C_3$ ($C_{337}:C_{168}$)

Order 2: $C_2$ ($C_{1011}:C_{84}$)

Order 1: $C_1$ ($C_{1011}:C_{168}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 169848: $C_{1011}:C_{168}$ ($C_1$)

Order 84924: $C_{1011}:C_{84}$ ($C_2$)

Order 56616: $C_{1011}:C_{56}$ ($C_3$), $C_{337}:C_{168}$ ($C_3$)

Order 42462: $C_{1011}:C_{42}$ ($C_4$)

Order 28308: $C_{1011}:C_{28}$ ($C_6$), $C_{337}:C_{84}$ ($C_6$)

Order 24264: $C_{1011}:C_{24}$ ($C_7$)

Order 21231: $C_{1011}:C_{21}$ ($C_8$)

Order 18872: $C_{337}:C_{56}$ ($C_3^2$)

Order 14154: $C_{337}:C_{42}$ ($C_{12}$), $C_{1011}:C_{14}$ ($C_{12}$)

Order 12132: $C_{1011}:C_{12}$ ($C_{14}$)

Order 9436: $C_{337}:C_{28}$ ($C_3\times C_6$)

Order 8088: $C_{1011}:C_8$ ($C_{21}$), $C_{337}:C_{24}$ ($C_{21}$)

Order 7077: $C_{1011}:C_7$ ($C_{24}$), $C_{337}:C_{21}$ ($C_{24}$)

Order 6066: $C_{1011}:C_6$ ($C_{28}$)

Order 4718: $C_{337}:C_{14}$ ($C_3\times C_{12}$)

Order 4044: $C_{1011}:C_4$ ($C_{42}$), $C_{337}:C_{12}$ ($C_{42}$)

Order 3033: $C_{1011}:C_3$ ($C_{56}$)

Order 2696: $C_{337}:C_8$ ($C_3\times C_{21}$)

Order 2359: $C_{337}:C_7$ ($C_3\times C_{24}$)

Order 2022: $C_{2022}$ ($C_{84}$), $C_{337}:C_6$ ($C_{84}$)

Order 1348: $C_{337}:C_4$ ($C_3\times C_{42}$)

Order 1011: $C_{1011}$ ($C_{168}$), $C_{337}:C_3$ ($C_{168}$)

Order 674: $C_{674}$ ($C_3\times C_{84}$)

Order 337: $C_{337}$ ($C_3\times C_{168}$)

Order 6: $C_6$ ($C_{337}:C_{84}$)

Order 3: $C_3$ ($C_{337}:C_{168}$)

Order 2: $C_2$ ($C_{1011}:C_{84}$)

Order 1: $C_1$ ($C_{1011}:C_{168}$)

Series

Derived series

$C_{1011}:C_{168}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{1011}:C_{168}$

$\rhd$

$C_{1011}:C_{84}$

$\rhd$

$C_{1011}:C_{42}$

$\rhd$

$C_{1011}:C_{21}$

$\rhd$

$C_{1011}:C_7$

$\rhd$

$C_{1011}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{1011}:C_{168}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $528 \times 528$ character table is not available for this group.

Rational character table

See the $44 \times 44$ rational character table.