Abstract group 150976.a: $C_{2696}:C_{56}$

• Label: 150976.a
• Order: \( 2^{6} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{3} \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $345$
• Trans deg.: $2696$
• Rank: $2$

Group information

Description:	$C_{2696}:C_{56}$
Order:	\(150976\)\(\medspace = 2^{6} \cdot 7 \cdot 337 \)
Exponent:	\(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \)
Automorphism group:	$C_{1348}.C_{336}.C_2^3$, of order \(3623424\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 6, $C_7$, $C_{337}$
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	2696
Elements	1	675	3372	2022	14832	6066	24264	97056	336	336	672	1344	150976
Conjugacy classes	1	3	12	6	48	18	72	288	6	6	12	24	496
Divisions	1	3	6	1	12	3	6	12	1	1	1	1	48
Autjugacy classes	1	2	4	6	8	12	24	48	1	1	1	1	109

Dimension	1	2	4	6	12	24	56	336	672	1344
Irr. complex chars.	448	0	0	0	0	0	48	0	0	0	496
Irr. rational chars.	4	6	12	4	6	12	0	2	1	1	48

Minimal presentations

Permutation degree:	$345$
Transitive degree:	$2696$
Rank:	$2$
Inequivalent generating pairs:	$2304$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	56	112	1344
Arbitrary	56	58	340

Constructions

Presentation:	$\langle a, b \mid a^{56}=b^{2696}=1, b^{a}=b^{713} \rangle$
Permutation group:	Degree $345$ $\langle(1,2,4,20,70,174,166,56,14,54,177,237,224,306,179,169,60,147,287,333,298,291,328,243,138,40,71,32,112,105,280,144,289,149,242,98,130,82,44,9,43,22,11,8,42,140,69,19,48,160,87,107,203,226,81,225,129,273,250,286,332,297,249,133,38,131,170,318,162,101,244,309,135,276,113,214,252,90,25,47,158,208,115,91,27,89,24,88,239,111,163,319,321,198,65,59,93,265,307,302,281,106,270,127,284,108,218,76,157,145,314,248,232,207,299,335,217,75,216,109,31,5,29,95,269,201,77,219,84,74,185,222,310,312,151,316,234,134,268,230,164,194,97,261,221,124,146,46,10,45,30,103,260,300,301,215,155,233,92,262,277,305,132,139,41,114,290,317,154,118,292,283,121,34,119,181,303,296,331,197,238,122,210,326,304,257,126,36,35,6,33,99,156,288,274,100,78,205,229,258,184,255,275,200,259,267,228,143,191,188,323,193,148,315,172,52,13,51,167,320,192,329,337,325,211,308,246,313,142,117,72,209,152,263,104,272,110,220,327,324,223,80,21,28,94,266,227,204,153,285,190,61,189,64,196,123,293,202,68,63,195,271,294,125,295,178,282,240,173,141,256,96,183,55,180,254,278,334,251,330,235,182,247,187,245,159,150,136,165,50,12,49,161,58,15,18,26,67,17,66,79,120,171,322,199,206,186,253,168,176,53,175,57,73,212,241,213,279,336,236,86,23,83,85,231,264,311,137,39,7,37,128,102,116,62,16,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 150 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 226 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_8$ $\, \times\, $ $(C_{337}:C_{56})$
Semidirect product:	$C_{2696}$ $\,\rtimes\,$ $C_{56}$	$(C_{2696}:C_8)$ $\,\rtimes\,$ $C_7$	$(C_{2696}:C_7)$ $\,\rtimes\,$ $C_8$	$(C_{337}:C_{56})$ $\,\rtimes\,$ $C_8$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{337}:C_{56})$ . $C_4$	$(C_8\times D_{337})$ . $C_{28}$	$(D_{337}:C_8)$ . $C_{28}$	$D_{674}$ . $(C_4\times C_{28})$	all 30

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

Homology

Abelianization:	$C_{8} \times C_{56} \simeq C_{8}^{2} \times C_{7}$
Schur multiplier:	$C_{8}$
Commutator length:	$1$

Subgroups

There are 23668 subgroups in 148 conjugacy classes, 78 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $C_{337}:C_{56}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_8\times C_{56}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{674}:C_{56}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2696}$	$G/\operatorname{Fit} \simeq$ $C_{56}$
Radical:	$R \simeq$ $C_{2696}:C_{56}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{674}$	$G/\operatorname{soc} \simeq$ $C_4\times C_{56}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8^2$
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

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 150976: $C_{2696}:C_{56}$

Order 75488: $C_{1348}:C_{56}$ x 2, $C_{2696}:C_{28}$

Order 37744: $C_{674}:C_{56}$ x 4, $C_{1348}:C_{28}$, $C_{2696}:C_{14}$, $D_{337}:C_{56}$

Order 21568: $C_{2696}:C_8$

Order 18872: $C_{337}:C_{56}$ x 8, $C_{674}:C_{28}$ x 2, $C_{337}:C_{56}$ x 2, $C_{337}:C_{56}$, $C_{2696}:C_7$, $C_{1348}:C_{14}$

Order 10784: $C_{1348}:C_8$ x 2, $C_{2696}:C_4$

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 4, $D_{337}:C_8$, $C_{1348}:C_4$, $C_8\times D_{337}$

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

Order 2696: $C_{337}:C_8$ x 8, $C_{674}:C_4$ x 2, $C_{337}:C_8$ x 2, $C_4\times D_{337}$, $C_{337}:C_8$, $C_{2696}$

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 448: $C_8\times C_{56}$

Order 337: $C_{337}$

Order 224: $C_4\times C_{56}$ x 3

Order 112: $C_2\times C_{56}$ x 6, $C_4\times C_{28}$

Order 64: $C_8^2$

Order 56: $C_{56}$ x 12, $C_2\times C_{28}$ x 3

Order 32: $C_4\times C_8$ x 3

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

Order 16: $C_2\times C_8$ x 6, $C_4^2$

Order 14: $C_{14}$ x 3

Order 8: $C_8$ x 12, $C_2\times C_4$ 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 150976: $C_{2696}:C_{56}$

Order 75488: $C_{2696}:C_{28}$, $C_{1348}:C_{56}$

Order 37744: $C_{1348}:C_{28}$, $C_{2696}:C_{14}$, $C_{674}:C_{56}$, $D_{337}:C_{56}$

Order 21568: $C_{2696}:C_8$

Order 18872: $C_{337}:C_{56}$, $C_{2696}:C_7$, $C_{674}:C_{28}$, $C_{337}:C_{56}$, $C_{1348}:C_{14}$, $C_{337}:C_{56}$

Order 10784: $C_{1348}:C_8$, $C_{2696}:C_4$

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

Order 5392: $C_{674}:C_8$, $D_{337}:C_8$, $C_{1348}:C_4$, $C_8\times D_{337}$

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

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

Order 2359: $C_{337}:C_7$

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

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

Order 448: $C_8\times C_{56}$

Order 337: $C_{337}$

Order 224: $C_4\times C_{56}$ x 2

Order 112: $C_2\times C_{56}$ x 3, $C_4\times C_{28}$

Order 64: $C_8^2$

Order 56: $C_{56}$ x 4, $C_2\times C_{28}$ x 2

Order 32: $C_4\times C_8$ x 2

Order 28: $C_{28}$ x 3, $C_2\times C_{14}$

Order 16: $C_2\times C_8$ x 3, $C_4^2$

Order 14: $C_{14}$ x 2

Order 8: $C_8$ x 4, $C_2\times C_4$ x 2

Order 7: $C_7$

Order 4: $C_4$ x 3, $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 150976: $C_{2696}:C_{56}$ ($C_1$)

Order 75488: $C_{1348}:C_{56}$ ($C_2$) x 2, $C_{2696}:C_{28}$ ($C_2$)

Order 37744: $C_{674}:C_{56}$ ($C_4$) x 4, $C_{1348}:C_{28}$ ($C_2^2$), $C_{2696}:C_{14}$ ($C_4$), $D_{337}:C_{56}$ ($C_4$)

Order 21568: $C_{2696}:C_8$ ($C_7$)

Order 18872: $C_{337}:C_{56}$ ($C_8$) x 8, $C_{674}:C_{28}$ ($C_2\times C_4$) x 2, $C_{337}:C_{56}$ ($C_8$) x 2, $C_{337}:C_{56}$ ($C_8$), $C_{2696}:C_7$ ($C_8$), $C_{1348}:C_{14}$ ($C_2\times C_4$)

Order 10784: $C_{1348}:C_8$ ($C_{14}$) x 2, $C_{2696}:C_4$ ($C_{14}$)

Order 9436: $C_{337}:C_{28}$ ($C_2\times C_8$) x 4, $C_{337}:C_{28}$ ($C_2\times C_8$), $C_{674}:C_{14}$ ($C_4^2$), $C_{1348}:C_7$ ($C_2\times C_8$)

Order 5392: $C_{674}:C_8$ ($C_{28}$) x 4, $D_{337}:C_8$ ($C_{28}$), $C_{1348}:C_4$ ($C_2\times C_{14}$), $C_8\times D_{337}$ ($C_{28}$)

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

Order 2696: $C_{337}:C_8$ ($C_{56}$) x 8, $C_{674}:C_4$ ($C_2\times C_{28}$) x 2, $C_{337}:C_8$ ($C_{56}$) x 2, $C_4\times D_{337}$ ($C_2\times C_{28}$), $C_{337}:C_8$ ($C_{56}$), $C_{2696}$ ($C_{56}$)

Order 2359: $C_{337}:C_7$ ($C_8^2$)

Order 1348: $C_{337}:C_4$ ($C_2\times C_{56}$) x 4, $D_{674}$ ($C_4\times C_{28}$), $C_{1348}$ ($C_2\times C_{56}$), $C_{337}:C_4$ ($C_2\times C_{56}$)

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

Order 337: $C_{337}$ ($C_8\times C_{56}$)

Order 8: $C_8$ ($C_{337}:C_{56}$)

Order 4: $C_4$ ($C_{674}:C_{56}$)

Order 2: $C_2$ ($C_{1348}:C_{56}$)

Order 1: $C_1$ ($C_{2696}:C_{56}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 150976: $C_{2696}:C_{56}$ ($C_1$)

Order 75488: $C_{2696}:C_{28}$ ($C_2$), $C_{1348}:C_{56}$ ($C_2$)

Order 37744: $C_{1348}:C_{28}$ ($C_2^2$), $C_{2696}:C_{14}$ ($C_4$), $C_{674}:C_{56}$ ($C_4$), $D_{337}:C_{56}$ ($C_4$)

Order 21568: $C_{2696}:C_8$ ($C_7$)

Order 18872: $C_{337}:C_{56}$ ($C_8$), $C_{2696}:C_7$ ($C_8$), $C_{674}:C_{28}$ ($C_2\times C_4$), $C_{337}:C_{56}$ ($C_8$), $C_{1348}:C_{14}$ ($C_2\times C_4$), $C_{337}:C_{56}$ ($C_8$)

Order 10784: $C_{1348}:C_8$ ($C_{14}$), $C_{2696}:C_4$ ($C_{14}$)

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

Order 5392: $C_{674}:C_8$ ($C_{28}$), $D_{337}:C_8$ ($C_{28}$), $C_{1348}:C_4$ ($C_2\times C_{14}$), $C_8\times D_{337}$ ($C_{28}$)

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

Order 2696: $C_4\times D_{337}$ ($C_2\times C_{28}$), $C_{337}:C_8$ ($C_{56}$), $C_{337}:C_8$ ($C_{56}$), $C_{674}:C_4$ ($C_2\times C_{28}$), $C_{337}:C_8$ ($C_{56}$), $C_{2696}$ ($C_{56}$)

Order 2359: $C_{337}:C_7$ ($C_8^2$)

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

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

Order 337: $C_{337}$ ($C_8\times C_{56}$)

Order 8: $C_8$ ($C_{337}:C_{56}$)

Order 4: $C_4$ ($C_{674}:C_{56}$)

Order 2: $C_2$ ($C_{1348}:C_{56}$)

Order 1: $C_1$ ($C_{2696}:C_{56}$)

Series

Derived series

$C_{2696}:C_{56}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{2696}:C_{56}$

$\rhd$

$C_{2696}:C_{28}$

$\rhd$

$C_{2696}:C_{14}$

$\rhd$

$C_{2696}:C_7$

$\rhd$

$C_{2696}$

$\rhd$

$C_{1348}$

$\rhd$

$C_{674}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{2696}:C_{56}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_8$

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

See the $496 \times 496$ 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 $48 \times 48$ rational character table.