Abstract group 75488.b: $C_{1348}:C_{56}$

• Label: 75488.b
• Order: \( 2^{5} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{3} \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $341$
• Trans deg.: $1348$
• Rank: $2$

Group information

Description:	$C_{1348}:C_{56}$
Order:	\(75488\)\(\medspace = 2^{5} \cdot 7 \cdot 337 \)
Exponent:	\(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \)
Automorphism group:	$C_{674}.C_{336}.C_2^2$, of order \(905856\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 5, $C_7$, $C_{337}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	8	14	28	56	337	674	1348
Elements	1	675	3372	2022	5392	6066	24264	32352	336	336	672	75488
Conjugacy classes	1	3	12	6	16	18	72	96	6	6	12	248
Divisions	1	3	6	1	4	3	6	4	1	1	1	31
Autjugacy classes	1	3	6	6	4	18	36	24	1	1	1	101

Dimension	1	2	4	6	12	24	56	336	672
Irr. complex chars.	224	0	0	0	0	0	24	0	0	248
Irr. rational chars.	4	6	4	4	6	4	0	2	1	31

Minimal presentations

Permutation degree:	$341$
Transitive degree:	$1348$
Rank:	$2$
Inequivalent generating pairs:	$2304$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	56	112	672
Arbitrary	56	58	338

Constructions

Presentation:	$\langle a, b \mid a^{56}=b^{1348}=1, b^{a}=b^{713} \rangle$
Permutation group:	Degree $341$ $\langle(1,2,4,26,108,220,74,25,17,82,203,337,195,124,98,245,91,244,294,169,332,255,330,305,151,40,150,42,154,300,188,334,321,314,212,71,13,70,209,80,207,68,12,61,192,164,191,60,9,53,121,157,44,156,122,48,54,178,268,185,262,326,229,81,16,79,222,306,325,270,310,143,35,142,295,273,336,265,102,22,101,261,248,141,34,5,37,144,271,291,292,129,217,309,198,136,31,135,130,285,114,186,57,8,55,179,316,275,116,260,100,21,99,259,138,32,137,307,170,211,266,103,215,84,231,234,115,243,90,242,247,92,246,223,297,311,187,77,167,313,296,320,235,284,113,110,278,148,38,147,83,230,263,204,225,286,279,256,96,213,302,181,56,180,69,208,241,89,240,120,27,118,267,105,159,45,158,290,140,33,139,287,318,163,264,200,172,49,171,64,196,331,249,250,324,184,282,317,322,153,41,152,299,329,162,133,30,131,125,28,123,253,183,177,52,176,67,206,269,228,280,224,95,205,194,63,10,62,193,254,274,327,238,87,18,86,233,252,304,134,189,93,146,127,298,182,174,335,333,216,165,75,15,73,202,109,201,66,11,65,197,312,319,277,258,97,257,308,214,303,323,166,47,6,43,155,301,199,236,210,232,85,59,145,219,72,218,190,58,19,50,173,226,239,88,23,104,161,46,160,107,24,106,168,328,272,315,237,283,112,78,221,76,111,281,289,117,288,227,276,128,29,126,36,132,119,293,175,51,7,14,39,149,251,94,20,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 85 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 150 & 0 \\ 0 & 173 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_4$ $\, \times\, $ $(C_{337}:C_{56})$
Semidirect product:	$C_{1348}$ $\,\rtimes\,$ $C_{56}$	$(C_{1348}:C_8)$ $\,\rtimes\,$ $C_7$	$(C_{1348}:C_7)$ $\,\rtimes\,$ $C_8$	$(C_{337}:C_{28})$ $\,\rtimes\,$ $C_8$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times D_{337})$ . $C_{28}$	$D_{674}$ . $(C_2\times C_{28})$	$D_{337}$ . $(C_4\times C_{28})$	$D_{337}$ . $(C_2\times C_{56})$	all 21

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

Homology

Abelianization:	$C_{4} \times C_{56} \simeq C_{4} \times C_{8} \times C_{7}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 13864 subgroups in 88 conjugacy classes, 47 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{337}:C_{56}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_4\times C_{56}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{674}:C_{56}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1348}$	$G/\operatorname{Fit} \simeq$ $C_{56}$
Radical:	$R \simeq$ $C_{1348}:C_{56}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{674}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{56}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_8$
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 75488: $C_{1348}:C_{56}$

Order 37744: $C_{674}:C_{56}$ x 2, $C_{1348}:C_{28}$

Order 18872: $C_{337}:C_{56}$ x 4, $C_{674}:C_{28}$ x 2, $C_{1348}:C_{14}$

Order 10784: $C_{1348}:C_8$

Order 9436: $C_{337}:C_{28}$ x 4, $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 5392: $C_{674}:C_8$ x 2, $C_{1348}:C_4$

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

Order 2696: $C_{337}:C_8$ x 4, $C_{674}:C_4$ x 2, $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 1348: $C_{337}:C_4$ x 4, $D_{674}$, $C_{1348}$, $C_{337}:C_4$

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

Order 337: $C_{337}$

Order 224: $C_4\times C_{56}$

Order 112: $C_2\times C_{56}$ x 2, $C_4\times C_{28}$

Order 56: $C_{56}$ x 4, $C_2\times C_{28}$ x 3

Order 32: $C_4\times C_8$

Order 28: $C_{28}$ x 6, $C_2\times C_{14}$

Order 16: $C_2\times C_8$ x 2, $C_4^2$

Order 14: $C_{14}$ x 3

Order 8: $C_8$ x 4, $C_2\times C_4$ x 3

Order 7: $C_7$

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 75488: $C_{1348}:C_{56}$

Order 37744: $C_{1348}:C_{28}$, $C_{674}:C_{56}$

Order 18872: $C_{674}:C_{28}$ x 2, $C_{337}:C_{56}$, $C_{1348}:C_{14}$

Order 10784: $C_{1348}:C_8$

Order 9436: $C_{337}:C_{28}$ x 2, $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 5392: $C_{674}:C_8$, $C_{1348}:C_4$

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

Order 2696: $C_{674}:C_4$ x 2, $C_4\times D_{337}$, $C_{337}:C_8$

Order 2359: $C_{337}:C_7$

Order 1348: $C_{337}:C_4$ x 2, $D_{674}$, $C_{1348}$, $C_{337}:C_4$

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

Order 337: $C_{337}$

Order 224: $C_4\times C_{56}$

Order 112: $C_2\times C_{56}$, $C_4\times C_{28}$

Order 56: $C_2\times C_{28}$ x 3, $C_{56}$

Order 32: $C_4\times C_8$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$

Order 16: $C_2\times C_8$, $C_4^2$

Order 14: $C_{14}$ x 3

Order 8: $C_2\times C_4$ x 3, $C_8$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 75488: $C_{1348}:C_{56}$ ($C_1$)

Order 37744: $C_{674}:C_{56}$ ($C_2$) x 2, $C_{1348}:C_{28}$ ($C_2$)

Order 18872: $C_{337}:C_{56}$ ($C_4$) x 4, $C_{674}:C_{28}$ ($C_2^2$), $C_{674}:C_{28}$ ($C_4$), $C_{1348}:C_{14}$ ($C_4$)

Order 10784: $C_{1348}:C_8$ ($C_7$)

Order 9436: $C_{337}:C_{28}$ ($C_2\times C_4$) x 2, $C_{337}:C_{28}$ ($C_8$) x 2, $C_{337}:C_{28}$ ($C_8$), $C_{674}:C_{14}$ ($C_2\times C_4$), $C_{1348}:C_7$ ($C_8$)

Order 5392: $C_{674}:C_8$ ($C_{14}$) x 2, $C_{1348}:C_4$ ($C_{14}$)

Order 4718: $C_{337}:C_{14}$ ($C_2\times C_8$), $C_{337}:C_{14}$ ($C_2\times C_8$), $C_{337}:C_{14}$ ($C_4^2$)

Order 2696: $C_{337}:C_8$ ($C_{28}$) x 4, $C_4\times D_{337}$ ($C_{28}$), $C_{674}:C_4$ ($C_2\times C_{14}$), $C_{674}:C_4$ ($C_{28}$)

Order 2359: $C_{337}:C_7$ ($C_4\times C_8$)

Order 1348: $C_{337}:C_4$ ($C_2\times C_{28}$) x 2, $C_{337}:C_4$ ($C_{56}$) x 2, $D_{674}$ ($C_2\times C_{28}$), $C_{1348}$ ($C_{56}$), $C_{337}:C_4$ ($C_{56}$)

Order 674: $C_{674}$ ($C_2\times C_{56}$), $D_{337}$ ($C_2\times C_{56}$), $D_{337}$ ($C_4\times C_{28}$)

Order 337: $C_{337}$ ($C_4\times C_{56}$)

Order 4: $C_4$ ($C_{337}:C_{56}$)

Order 2: $C_2$ ($C_{674}:C_{56}$)

Order 1: $C_1$ ($C_{1348}:C_{56}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 75488: $C_{1348}:C_{56}$ ($C_1$)

Order 37744: $C_{1348}:C_{28}$ ($C_2$), $C_{674}:C_{56}$ ($C_2$)

Order 18872: $C_{674}:C_{28}$ ($C_2^2$), $C_{674}:C_{28}$ ($C_4$), $C_{337}:C_{56}$ ($C_4$), $C_{1348}:C_{14}$ ($C_4$)

Order 10784: $C_{1348}:C_8$ ($C_7$)

Order 9436: $C_{337}:C_{28}$ ($C_8$), $C_{674}:C_{14}$ ($C_2\times C_4$), $C_{1348}:C_7$ ($C_8$), $C_{337}:C_{28}$ ($C_2\times C_4$), $C_{337}:C_{28}$ ($C_8$)

Order 5392: $C_{674}:C_8$ ($C_{14}$), $C_{1348}:C_4$ ($C_{14}$)

Order 4718: $C_{337}:C_{14}$ ($C_2\times C_8$), $C_{337}:C_{14}$ ($C_2\times C_8$), $C_{337}:C_{14}$ ($C_4^2$)

Order 2696: $C_4\times D_{337}$ ($C_{28}$), $C_{337}:C_8$ ($C_{28}$), $C_{674}:C_4$ ($C_2\times C_{14}$), $C_{674}:C_4$ ($C_{28}$)

Order 2359: $C_{337}:C_7$ ($C_4\times C_8$)

Order 1348: $D_{674}$ ($C_2\times C_{28}$), $C_{337}:C_4$ ($C_2\times C_{28}$), $C_{337}:C_4$ ($C_{56}$), $C_{1348}$ ($C_{56}$), $C_{337}:C_4$ ($C_{56}$)

Order 674: $C_{674}$ ($C_2\times C_{56}$), $D_{337}$ ($C_2\times C_{56}$), $D_{337}$ ($C_4\times C_{28}$)

Order 337: $C_{337}$ ($C_4\times C_{56}$)

Order 4: $C_4$ ($C_{337}:C_{56}$)

Order 2: $C_2$ ($C_{674}:C_{56}$)

Order 1: $C_1$ ($C_{1348}:C_{56}$)

Series

Derived series

$C_{1348}:C_{56}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{1348}:C_{56}$

$\rhd$

$C_{1348}:C_{28}$

$\rhd$

$C_{1348}:C_{14}$

$\rhd$

$C_{1348}:C_7$

$\rhd$

$C_{1348}$

$\rhd$

$C_{674}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{1348}:C_{56}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $248 \times 248$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $31 \times 31$ rational character table.