Abstract group 61152.a: $C_{56}.\PSL(2,13)$

• Label: 61152.a
• Order: \( 2^{5} \cdot 3 \cdot 7^{2} \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $119$
• Trans deg.: $784$
• Rank: $2$

Group information

Description:	$C_{56}.\PSL(2,13)$
Order:	\(61152\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{2} \cdot 13 \)
Exponent:	\(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Automorphism group:	$C_2^2\times C_6\times \PSL(2,13).C_2$, of order \(52416\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \cdot 13 \)
Composition factors:	$C_2$ x 3, $C_7$, $\PSL(2,13)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	13	14	21	24	26	28	42	52	56	84	91	104	168	182	364	728
Elements	1	183	182	184	546	3282	368	728	168	4374	1092	1456	168	7656	3276	336	15312	4368	1008	672	8736	1008	2016	4032	61152
Conjugacy classes	1	2	1	3	3	27	6	4	2	33	6	8	2	60	18	4	120	24	12	8	48	12	24	48	476
Divisions	1	2	1	2	2	5	2	2	1	6	1	2	1	6	2	1	6	2	1	1	2	1	1	1	52
Autjugacy classes	1	2	1	2	2	7	2	3	1	8	1	2	1	8	2	1	8	3	1	1	2	1	1	1	62

Dimension	1	2	6	7	12	13	14	26	28	36	48	56	72	78	84	112	144	156	168	288	336	672
Irr. complex chars.	28	0	56	56	168	28	140	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	476
Irr. rational chars.	2	1	2	0	1	2	6	1	3	2	1	1	7	2	6	1	4	1	3	4	1	1	52

Minimal presentations

Permutation degree:	$119$
Transitive degree:	$784$
Rank:	$2$
Inequivalent generating pairs:	$47520$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	288
Arbitrary	6	12	54

Constructions

Permutation group:	Degree $119$ $\langle(1,2,4,7)(3,17,19,22)(5,12,9,15)(6,13,25,29)(8,34,26,35)(10,16,27,33)(11,39,31,42) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{25} & \alpha^{126} \\ \alpha^{164} & \alpha^{109} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{146} & \alpha^{96} \\ \alpha^{60} & \alpha^{84} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{132} & 0 \\ 0 & \alpha^{132} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{150} & 0 \\ 0 & \alpha^{150} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{120} & 0 \\ 0 & \alpha^{120} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{159} & 0 \\ 0 & \alpha^{159} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{169}) = \GL_{2}(\F_{13}[\alpha]/(\alpha^{2} + 12 \alpha + 2))$
Direct product:	$C_7$ $\, \times\, $ $(C_8.\PSL(2,13))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\SL(2,13)$ . $C_{28}$	$C_{56}$ . $\PSL(2,13)$	$(C_7\times \SL(2,13))$ . $C_4$	$(\SL(2,13):C_{14})$ . $C_2$	all 10
Aut. group:	$\Aut(C_2^5.C_2^6:C_{10})$	$\Aut(C_2^5.C_2^6:C_{10})$

Elements of the group are displayed as matrices in $\GL_{2}(\F_{169})$.

Homology

Abelianization:	$C_{28} \simeq C_{4} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 13864 subgroups in 182 conjugacy classes, 14 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{56}$	$G/Z \simeq$ $\PSL(2,13)$
Commutator:	$G' \simeq$ $\SL(2,13)$	$G/G' \simeq$ $C_{28}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{14}.\PSL(2,13)$
Fitting:	$\operatorname{Fit} \simeq$ $C_{56}$	$G/\operatorname{Fit} \simeq$ $\PSL(2,13)$
Radical:	$R \simeq$ $C_{56}$	$G/R \simeq$ $\PSL(2,13)$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_4.\PSL(2,13)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 61152: $C_{56}.\PSL(2,13)$

Order 30576: $\SL(2,13):C_{14}$

Order 15288: $C_7\times \SL(2,13)$

Order 8736: $C_8.\PSL(2,13)$

Order 4368: $C_{104}:C_{42}$, $\SL(2,13):C_2$

Order 2184: $C_{13}:C_{168}$, $C_{364}:C_6$, $C_{13}:C_{168}$, $\SL(2,13)$

Order 1456: $D_{13}\times C_{56}$

Order 1092: $C_{26}:C_{42}$, $C_{13}:C_{84}$, $C_{13}:C_{84}$

Order 784: $D_7\times C_{56}$

Order 728: $C_{728}$, $C_{13}:C_{56}$, $D_{13}\times C_{28}$

Order 672: $D_{12}.C_{28}$, $C_{56}.A_4$

Order 624: $C_{104}:C_6$

Order 546: $C_{13}:C_{42}$ x 2, $C_{13}:C_{42}$

Order 392: $D_7\times C_{28}$, $C_7\times C_{56}$, $C_7:C_{56}$

Order 364: $C_7\times D_{26}$, $C_{364}$, $C_{13}:C_{28}$

Order 336: $C_{168}:C_2$ x 2, $S_3\times C_{56}$ x 2, $C_{12}.C_{28}$, $D_{12}:C_{14}$, $\SL(2,3):C_{14}$, $C_2\times C_{168}$

Order 312: $C_{52}:C_6$, $C_{13}:C_{24}$, $C_{13}:C_{24}$

Order 273: $C_{13}:C_{21}$

Order 224: $\OD_{16}:C_{14}$

Order 208: $C_8\times D_{13}$

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$, $C_7\times D_{14}$

Order 182: $C_7\times D_{13}$ x 2, $C_{182}$

Order 168: $C_{168}$ x 2, $D_6:C_{14}$ x 2, $S_3\times C_{28}$ x 2, $C_3:C_{56}$ x 2, $C_2\times C_{84}$, $C_7\times D_{12}$, $C_{21}:Q_8$, $C_7\times \SL(2,3)$

Order 156: $C_{26}:C_6$, $C_{13}:C_{12}$, $C_{13}:C_{12}$

Order 112: $D_4:C_{14}$, $C_8\times D_7$, $C_7\times \OD_{16}$, $C_2\times C_{56}$

Order 104: $C_4\times D_{13}$, $C_{104}$, $C_{13}:C_8$

Order 98: $C_7\times D_7$ x 2, $C_7\times C_{14}$

Order 96: $C_8.A_4$, $C_8.D_6$

Order 91: $C_{91}$

Order 84: $C_{84}$ x 2, $C_3:C_{28}$ x 2, $S_3\times C_{14}$ x 2, $C_2\times C_{42}$

Order 78: $C_{13}:C_6$ x 2, $C_{13}:C_6$

Order 56: $C_{56}$ x 6, $C_7\times D_4$, $C_2\times C_{28}$, $C_4\times D_7$, $C_7\times Q_8$, $C_7:C_8$

Order 52: $D_{26}$, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_{12}:C_2$, $\SL(2,3):C_2$, $C_2\times C_{24}$, $C_3:\OD_{16}$

Order 42: $C_{42}$ x 2, $S_3\times C_7$ x 2

Order 39: $C_{13}:C_3$

Order 32: $\OD_{16}:C_2$

Order 28: $C_{28}$ x 6, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 26: $D_{13}$ x 2, $C_{26}$

Order 24: $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$, $\SL(2,3)$

Order 21: $C_{21}$

Order 16: $\OD_{16}$, $C_2\times C_8$, $D_4:C_2$

Order 14: $C_{14}$ x 6, $D_7$ x 2

Order 13: $C_{13}$

Order 12: $D_6$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $Q_8$, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 5

Order 6: $C_6$ x 2, $S_3$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 61152: $C_{56}.\PSL(2,13)$

Order 30576: $\SL(2,13):C_{14}$

Order 15288: $C_7\times \SL(2,13)$

Order 8736: $C_8.\PSL(2,13)$

Order 4368: $C_{104}:C_{42}$, $\SL(2,13):C_2$

Order 2184: $C_{13}:C_{168}$, $C_{364}:C_6$, $C_{13}:C_{168}$, $\SL(2,13)$

Order 1456: $D_{13}\times C_{56}$

Order 1092: $C_{26}:C_{42}$, $C_{13}:C_{84}$, $C_{13}:C_{84}$

Order 784: $D_7\times C_{56}$

Order 728: $C_{728}$, $C_{13}:C_{56}$, $D_{13}\times C_{28}$

Order 672: $D_{12}.C_{28}$, $C_{56}.A_4$

Order 624: $C_{104}:C_6$

Order 546: $C_{13}:C_{42}$, $C_{13}:C_{42}$

Order 392: $D_7\times C_{28}$, $C_7\times C_{56}$, $C_7:C_{56}$

Order 364: $C_7\times D_{26}$, $C_{364}$, $C_{13}:C_{28}$

Order 336: $C_{12}.C_{28}$, $C_{168}:C_2$, $S_3\times C_{56}$, $D_{12}:C_{14}$, $\SL(2,3):C_{14}$, $C_2\times C_{168}$

Order 312: $C_{52}:C_6$, $C_{13}:C_{24}$, $C_{13}:C_{24}$

Order 273: $C_{13}:C_{21}$

Order 224: $\OD_{16}:C_{14}$

Order 208: $C_8\times D_{13}$

Order 196: $C_7\times C_{28}$, $C_7:C_{28}$, $C_7\times D_{14}$

Order 182: $C_{182}$, $C_7\times D_{13}$

Order 168: $C_{168}$ x 2, $C_2\times C_{84}$, $D_6:C_{14}$, $C_7\times D_{12}$, $S_3\times C_{28}$, $C_3:C_{56}$, $C_{21}:Q_8$, $C_7\times \SL(2,3)$

Order 156: $C_{26}:C_6$, $C_{13}:C_{12}$, $C_{13}:C_{12}$

Order 112: $D_4:C_{14}$, $C_8\times D_7$, $C_7\times \OD_{16}$, $C_2\times C_{56}$

Order 104: $C_4\times D_{13}$, $C_{104}$, $C_{13}:C_8$

Order 98: $C_7\times C_{14}$, $C_7\times D_7$

Order 96: $C_8.A_4$, $C_8.D_6$

Order 91: $C_{91}$

Order 84: $C_{84}$ x 2, $C_3:C_{28}$, $C_2\times C_{42}$, $S_3\times C_{14}$

Order 78: $C_{13}:C_6$, $C_{13}:C_6$

Order 56: $C_{56}$ x 4, $C_7\times D_4$, $C_2\times C_{28}$, $C_4\times D_7$, $C_7\times Q_8$, $C_7:C_8$

Order 52: $D_{26}$, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_{24}:C_2$, $S_3\times C_8$, $D_{12}:C_2$, $\SL(2,3):C_2$, $C_2\times C_{24}$, $C_3:\OD_{16}$

Order 42: $C_{42}$ x 2, $S_3\times C_7$

Order 39: $C_{13}:C_3$

Order 32: $\OD_{16}:C_2$

Order 28: $C_{28}$ x 4, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 26: $C_{26}$, $D_{13}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $\SL(2,3)$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $\OD_{16}$, $C_2\times C_8$, $D_4:C_2$

Order 14: $C_{14}$ x 4, $D_7$

Order 13: $C_{13}$

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 8: $C_8$ x 2, $Q_8$, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 3

Order 6: $C_6$ x 2, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 61152: $C_{56}.\PSL(2,13)$ ($C_1$)

Order 30576: $\SL(2,13):C_{14}$ ($C_2$)

Order 15288: $C_7\times \SL(2,13)$ ($C_4$)

Order 8736: $C_8.\PSL(2,13)$ ($C_7$)

Order 4368: $\SL(2,13):C_2$ ($C_{14}$)

Order 2184: $\SL(2,13)$ ($C_{28}$)

Order 56: $C_{56}$ ($\PSL(2,13)$)

Order 28: $C_{28}$ ($C_2\times \PSL(2,13)$)

Order 14: $C_{14}$ ($C_4.\PSL(2,13)$)

Order 8: $C_8$ ($C_7\times \PSL(2,13)$)

Order 7: $C_7$ ($C_8.\PSL(2,13)$)

Order 4: $C_4$ ($C_{14}.\PSL(2,13)$)

Order 2: $C_2$ ($C_{28}.\PSL(2,13)$)

Order 1: $C_1$ ($C_{56}.\PSL(2,13)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 61152: $C_{56}.\PSL(2,13)$ ($C_1$)

Order 30576: $\SL(2,13):C_{14}$ ($C_2$)

Order 15288: $C_7\times \SL(2,13)$ ($C_4$)

Order 8736: $C_8.\PSL(2,13)$ ($C_7$)

Order 4368: $\SL(2,13):C_2$ ($C_{14}$)

Order 2184: $\SL(2,13)$ ($C_{28}$)

Order 56: $C_{56}$ ($\PSL(2,13)$)

Order 28: $C_{28}$ ($C_2\times \PSL(2,13)$)

Order 14: $C_{14}$ ($C_4.\PSL(2,13)$)

Order 8: $C_8$ ($C_7\times \PSL(2,13)$)

Order 7: $C_7$ ($C_8.\PSL(2,13)$)

Order 4: $C_4$ ($C_{14}.\PSL(2,13)$)

Order 2: $C_2$ ($C_{28}.\PSL(2,13)$)

Order 1: $C_1$ ($C_{56}.\PSL(2,13)$)

Series

Derived series

$C_{56}.\PSL(2,13)$

$\rhd$

$\SL(2,13)$

Chief series

$C_{56}.\PSL(2,13)$

$\rhd$

$C_{56}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{56}.\PSL(2,13)$

$\rhd$

$\SL(2,13)$

Upper central series

$C_1$

$\lhd$

$C_{56}$

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 $476 \times 476$ 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 $52 \times 52$ rational character table.