Abstract group 8736.c: $C_8.\PSL(2,13)$

• Label: 8736.c
• Order: \( 2^{5} \cdot 3 \cdot 7 \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $112$
• Trans deg.: $112$
• Rank: $2$

Group information

Description:	$C_8.\PSL(2,13)$
Order:	\(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \)
Exponent:	\(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Automorphism group:	$C_2^2\times \PSL(2,13).C_2$, of order \(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \)
Composition factors:	$C_2$ x 3, $\PSL(2,13)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	13	14	24	26	28	52	56	104
Elements	1	183	182	184	546	468	368	728	168	468	1456	168	936	336	1872	672	8736
Conjugacy classes	1	2	1	3	3	3	6	4	2	3	8	2	6	4	12	8	68
Divisions	1	2	1	2	2	1	2	2	1	1	2	1	1	1	1	1	22
Autjugacy classes	1	2	1	2	2	3	2	3	1	3	2	1	3	1	3	1	31

Dimension	1	2	6	7	12	13	14	26	28	36	48	56	72	112	144
Irr. complex chars.	4	0	8	8	24	4	20	0	0	0	0	0	0	0	0	68
Irr. rational chars.	2	1	0	0	0	2	6	1	3	2	1	1	1	1	1	22

Minimal presentations

Permutation degree:	$112$
Transitive degree:	$112$
Rank:	$2$
Inequivalent generating pairs:	$5940$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	48
Arbitrary	6	12	48

Constructions

Permutation group:	Degree $112$ $\langle(1,2,6,9,3,5,10,7)(4,11,14,22,8,12,19,17)(13,29,41,28,23,34,44,21)(15,30,40,75,24,35,53,52) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{63} & 0 \\ 0 & \alpha^{63} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{84} & 0 \\ 0 & \alpha^{84} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{73} & \alpha^{6} \\ \alpha^{44} & \alpha^{157} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{2} & \alpha^{120} \\ \alpha^{84} & \alpha^{108} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{126} & 0 \\ 0 & \alpha^{126} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{169}) = \GL_{2}(\F_{13}[\alpha]/(\alpha^{2} + 12 \alpha + 2))$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$\SL(2,13)$ . $C_4$	$C_8$ . $\PSL(2,13)$	$(\SL(2,13):C_2)$ . $C_2$	$C_4$ . $(C_2\times \PSL(2,13))$	all 5
Aut. group:	$\Aut(C_2^5.C_2^6:C_{10})$

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 5996 subgroups in 85 conjugacy classes, 7 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $\PSL(2,13)$
Commutator:	$G' \simeq$ $\SL(2,13)$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times \PSL(2,13)$
Fitting:	$\operatorname{Fit} \simeq$ $C_8$	$G/\operatorname{Fit} \simeq$ $\PSL(2,13)$
Radical:	$R \simeq$ $C_8$	$G/R \simeq$ $\PSL(2,13)$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_4.\PSL(2,13)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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

Order 4368: $\SL(2,13):C_2$

Order 2184: $\SL(2,13)$

Order 624: $C_{104}:C_6$

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

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

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

Order 112: $C_8\times D_7$

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

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

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

Order 56: $C_4\times D_7$, $C_{56}$, $C_7:C_8$

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

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 39: $C_{13}:C_3$

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

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

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

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

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: $C_7$

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

Order 4368: $\SL(2,13):C_2$

Order 2184: $\SL(2,13)$

Order 624: $C_{104}:C_6$

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

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

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

Order 112: $C_8\times D_7$

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

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

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

Order 56: $C_4\times D_7$, $C_{56}$, $C_7:C_8$

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

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 39: $C_{13}:C_3$

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

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

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

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

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: $C_7$

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 8736: $C_8.\PSL(2,13)$ ($C_1$)

Order 4368: $\SL(2,13):C_2$ ($C_2$)

Order 2184: $\SL(2,13)$ ($C_4$)

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

Order 4: $C_4$ ($C_2\times \PSL(2,13)$)

Order 2: $C_2$ ($C_4.\PSL(2,13)$)

Order 1: $C_1$ ($C_8.\PSL(2,13)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8736: $C_8.\PSL(2,13)$ ($C_1$)

Order 4368: $\SL(2,13):C_2$ ($C_2$)

Order 2184: $\SL(2,13)$ ($C_4$)

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

Order 4: $C_4$ ($C_2\times \PSL(2,13)$)

Order 2: $C_2$ ($C_4.\PSL(2,13)$)

Order 1: $C_1$ ($C_8.\PSL(2,13)$)

Series

Derived series	$C_8.\PSL(2,13)$	$\rhd$	$\SL(2,13)$
Chief series	$C_8.\PSL(2,13)$	$\rhd$	$C_8$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_8.\PSL(2,13)$	$\rhd$	$\SL(2,13)$
Upper central series	$C_1$	$\lhd$	$C_8$

Supergroups

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

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

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.