Abstract group 122304.a: $C_{56}.\PGL(2,13)$

• Label: 122304.a
• Order: \( 2^{6} \cdot 3 \cdot 7^{2} \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $119$
• Trans deg.: $784$
• Rank: $2$

Group information

Description:	$C_{56}.\PGL(2,13)$
Order:	\(122304\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{2} \cdot 13 \)
Exponent:	\(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Automorphism group:	$C_2^3\times C_6\times \PSL(2,13).C_2$, of order \(104832\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \cdot 13 \)
Composition factors:	$C_2$ x 4, $C_7$, $\PSL(2,13)$
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	339	182	1068	546	3282	1408	2184	168	11862	1092	2912	168	19512	3276	336	34656	13104	1008	672	17472	1008	2016	4032	122304
Conjugacy classes	1	3	1	8	3	27	12	12	1	81	6	16	1	132	18	2	240	72	6	4	96	6	12	24	784
Divisions	1	3	1	5	2	5	5	4	1	14	1	4	1	13	2	1	13	4	1	1	4	1	1	1	89
Autjugacy classes	1	3	1	4	2	7	5	5	1	15	1	6	1	16	2	1	17	5	1	1	6	1	1	1	104

Dimension	1	2	6	12	13	14	26	28	36	48	56	72	78	84	112	144	156	168	288	336	672
Irr. complex chars.	56	0	0	364	56	308	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	784
Irr. rational chars.	4	2	4	2	4	6	2	5	4	1	3	14	4	6	2	6	2	5	8	3	2	89

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	12	not computed	288
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $119$ $\langle(1,2)(3,13)(4,9)(5,14)(6,15)(7,16)(8,17)(10,18)(11,19)(12,20)(21,55)(22,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{132} & 0 \\ 0 & \alpha^{132} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{17} & \alpha^{6} \\ \alpha^{64} & \alpha^{6} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{150} & 0 \\ 0 & \alpha^{150} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{120} & 0 \\ 0 & \alpha^{120} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{36} & \alpha^{158} \\ \alpha^{72} & \alpha^{12} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{132} & \alpha^{2} \\ \alpha^{60} & \alpha^{89} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{159} & 0 \\ 0 & \alpha^{159} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{169}) = \GL_{2}(\F_{13}[\alpha]/(\alpha^{2} + 12 \alpha + 2))$
Direct product:	$C_7$ $\, \times\, $ $(C_8.\PGL(2,13))$
Semidirect product:	$\Unitary(2,13)$ $\,\rtimes\,$ $C_4$	$(\SL(2,13):C_{28})$ $\,\rtimes\,$ $C_2$	$(\SL(2,13):C_4)$ $\,\rtimes\,$ $C_{14}$	$(\SL(2,13):C_2)$ $\,\rtimes\,$ $C_{28}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{56}$ . $\PGL(2,13)$	$\SL(2,13)$ . $(C_2\times C_{28})$	$(C_{28}.\PGL(2,13))$ . $C_2$	$(C_4.\PGL(2,13))$ . $C_{14}$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{28} \simeq C_{2} \times C_{4} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 43012 subgroups in 411 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{56}$	$G/Z \simeq$ $\PGL(2,13)$
Commutator:	$G' \simeq$ $\SL(2,13)$	$G/G' \simeq$ $C_2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{14}.\PSL(2,13).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{56}$	$G/\operatorname{Fit} \simeq$ $\PGL(2,13)$
Radical:	$R \simeq$ $C_{56}$	$G/R \simeq$ $\PGL(2,13)$
Socle:	$\operatorname{soc} \simeq$ $C_{14}$	$G/\operatorname{soc} \simeq$ $C_4.\PSL(2,13).C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_4$
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 122304: $C_{56}.\PGL(2,13)$

Order 61152: $\SL(2,13):C_{28}$, $C_{28}.\PGL(2,13)$, $C_{56}.\PSL(2,13)$

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

Order 17472: $C_8.\PGL(2,13)$

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

Order 8736: $C_4.\PGL(2,13)$, $C_8.\PSL(2,13)$, $C_{56}\times F_{13}$, $\SL(2,13):C_4$

Order 4368: $D_{13}:C_{168}$, $C_{28}\times F_{13}$, $C_{104}:C_{42}$, $\SL(2,13):C_2$, $\SL(2,13):C_2$, $C_2.\PGL(2,13)$

Order 2912: $C_{104}:C_{28}$

Order 2184: $C_{13}:C_{168}$ x 2, $C_{14}\times F_{13}$ x 2, $C_{13}:C_{168}$, $C_{364}:C_6$, $C_{13}:C_{168}$, $\SL(2,13)$

Order 1568: $C_{56}.D_{14}$

Order 1456: $C_{364}:C_4$, $D_{13}:C_{56}$, $D_{13}\times C_{56}$

Order 1344: $\GL(2,3):C_{28}$, $D_{24}:C_{28}$

Order 1248: $C_8\times F_{13}$

Order 1092: $C_7\times F_{13}$ x 4, $C_{26}:C_{42}$, $C_{13}:C_{84}$, $C_{13}:C_{84}$

Order 784: $C_{56}:C_{14}$ x 2, $D_7\times C_{56}$ x 2, $C_{14}\times C_{56}$, $C_{28}.C_{28}$, $D_{28}:C_{14}$

Order 728: $C_{13}:C_{56}$ x 2, $C_{26}:C_{28}$ x 2, $C_{728}$, $C_{13}:C_{56}$, $D_{13}\times C_{28}$

Order 672: $D_{12}.C_{28}$ x 2, $D_{12}:C_{28}$ x 2, $D_{24}:C_{14}$, $C_{56}.A_4$, $\SL(2,3):C_{28}$, $C_4\times C_{168}$, $C_{24}.C_{28}$, $\GL(2,3):C_{14}$

Order 624: $C_{104}:C_6$, $C_4\times F_{13}$, $D_{13}:C_{24}$

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

Order 448: $D_8:C_{28}$

Order 416: $C_{104}:C_4$

Order 392: $C_{14}\wr C_2$ x 2, $D_7\times C_{28}$ x 2, $C_7\times C_{56}$ x 2, $C_7:C_{56}$ x 2, $C_{14}\times C_{28}$, $C_7\times D_{28}$, $C_7^2:Q_8$

Order 364: $C_{13}:C_{28}$ x 4, $C_7\times D_{26}$, $C_{364}$, $C_{13}:C_{28}$

Order 336: $C_{12}.C_{28}$ x 2, $C_{168}:C_2$ x 2, $C_{168}:C_2$ x 2, $S_3\times C_{56}$ x 2, $D_{12}:C_{14}$ x 2, $C_2\times C_{168}$ x 2, $C_{21}:Q_{16}$, $C_7\times D_{24}$, $\SL(2,3):C_{14}$, $C_7\times \GL(2,3)$, $C_{14}.S_4$, $C_4\times C_{84}$

Order 312: $C_{13}:C_{24}$ x 2, $C_2\times F_{13}$ x 2, $C_{52}:C_6$, $C_{13}:C_{24}$, $C_{13}:C_{24}$

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

Order 224: $D_4:C_{28}$ x 2, $\OD_{16}:C_{14}$ x 2, $C_8.D_{14}$, $C_8.C_{28}$, $C_4\times C_{56}$, $D_8:C_{14}$

Order 208: $C_8\times D_{13}$, $C_{52}:C_4$, $D_{13}:C_8$

Order 196: $C_7\times C_{28}$ x 2, $C_7:C_{28}$ x 2, $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 192: $\GL(2,3):C_4$, $D_{24}:C_4$

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

Order 168: $C_{168}$ x 4, $C_2\times C_{84}$ x 2, $D_6:C_{14}$ x 2, $C_7\times D_{12}$ x 2, $S_3\times C_{28}$ x 2, $C_3:C_{56}$ x 2, $C_{21}:Q_8$ x 2, $C_7\times \SL(2,3)$

Order 156: $F_{13}$ x 4, $C_{26}:C_6$, $C_{13}:C_{12}$, $C_{13}:C_{12}$

Order 112: $C_2\times C_{56}$ x 7, $C_7\times \OD_{16}$ x 3, $D_4:C_{14}$ x 2, $C_{56}:C_2$ x 2, $C_8\times D_7$ x 2, $C_7\times \SD_{16}$ x 2, $C_7:\OD_{16}$, $D_{28}:C_2$, $C_7\times Q_{16}$, $C_7\times D_8$, $C_4\times C_{28}$

Order 104: $C_{13}:C_8$ x 2, $C_{26}:C_4$ x 2, $C_4\times D_{13}$, $C_{104}$, $C_{13}:C_8$

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

Order 96: $D_{12}:C_4$ x 2, $C_8.D_6$ x 2, $C_8.A_4$, $\Unitary(2,3)$, $C_4\times C_{24}$, $C_{24}.C_4$, $\GL(2,3):C_2$, $D_{24}:C_2$

Order 91: $C_{91}$

Order 84: $C_{84}$ x 4, $C_3:C_{28}$ 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 64: $D_8:C_4$

Order 56: $C_{56}$ x 13, $C_2\times C_{28}$ x 7, $C_7\times D_4$ x 3, $C_7:D_4$ x 2, $C_4\times D_7$ x 2, $C_7\times Q_8$ x 2, $C_7:C_8$ x 2, $D_{28}$, $C_7:Q_8$

Order 52: $C_{13}:C_4$ x 4, $D_{26}$, $C_{52}$, $C_{13}:C_4$

Order 49: $C_7^2$

Order 48: $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_{12}:C_2$ x 2, $C_2\times C_{24}$ x 2, $C_3:\OD_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_2.S_4$, $C_4\times C_{12}$

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

Order 39: $C_{13}:C_3$

Order 32: $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 28: $C_{28}$ x 13, $C_2\times C_{14}$ x 6, $D_{14}$ x 2, $C_7:C_4$ x 2

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

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 2, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $\SL(2,3)$

Order 21: $C_{21}$

Order 16: $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

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

Order 13: $C_{13}$

Order 12: $C_{12}$ x 4, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 7: $C_7$ x 5

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

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 122304: $C_{56}.\PGL(2,13)$

Order 61152: $\SL(2,13):C_{28}$, $C_{28}.\PGL(2,13)$, $C_{56}.\PSL(2,13)$

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

Order 17472: $C_8.\PGL(2,13)$

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

Order 8736: $C_4.\PGL(2,13)$, $C_8.\PSL(2,13)$, $C_{56}\times F_{13}$, $\SL(2,13):C_4$

Order 4368: $D_{13}:C_{168}$, $C_{28}\times F_{13}$, $C_{104}:C_{42}$, $\SL(2,13):C_2$, $\SL(2,13):C_2$, $C_2.\PGL(2,13)$

Order 2912: $C_{104}:C_{28}$

Order 2184: $C_{13}:C_{168}$ x 2, $C_{13}:C_{168}$, $C_{364}:C_6$, $C_{14}\times F_{13}$, $C_{13}:C_{168}$, $\SL(2,13)$

Order 1568: $C_{56}.D_{14}$

Order 1456: $C_{364}:C_4$, $D_{13}:C_{56}$, $D_{13}\times C_{56}$

Order 1344: $\GL(2,3):C_{28}$, $D_{24}:C_{28}$

Order 1248: $C_8\times F_{13}$

Order 1092: $C_{26}:C_{42}$, $C_7\times F_{13}$, $C_{13}:C_{84}$, $C_{13}:C_{84}$

Order 784: $C_{56}:C_{14}$ x 2, $D_7\times C_{56}$ x 2, $C_{14}\times C_{56}$, $C_{28}.C_{28}$, $D_{28}:C_{14}$

Order 728: $C_{13}:C_{56}$ x 2, $C_{728}$, $C_{26}:C_{28}$, $C_{13}:C_{56}$, $D_{13}\times C_{28}$

Order 672: $D_{12}.C_{28}$ x 2, $D_{12}:C_{28}$ x 2, $D_{24}:C_{14}$, $C_{56}.A_4$, $\SL(2,3):C_{28}$, $C_4\times C_{168}$, $C_{24}.C_{28}$, $\GL(2,3):C_{14}$

Order 624: $C_{104}:C_6$, $C_4\times F_{13}$, $D_{13}:C_{24}$

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

Order 448: $D_8:C_{28}$

Order 416: $C_{104}:C_4$

Order 392: $C_{14}\wr C_2$ x 2, $D_7\times C_{28}$ x 2, $C_7\times C_{56}$ x 2, $C_7:C_{56}$ x 2, $C_{14}\times C_{28}$, $C_7\times D_{28}$, $C_7^2:Q_8$

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

Order 336: $C_{12}.C_{28}$ x 2, $C_{168}:C_2$ x 2, $C_{168}:C_2$ x 2, $S_3\times C_{56}$ x 2, $D_{12}:C_{14}$ x 2, $C_2\times C_{168}$ x 2, $C_{21}:Q_{16}$, $C_7\times D_{24}$, $\SL(2,3):C_{14}$, $C_7\times \GL(2,3)$, $C_{14}.S_4$, $C_4\times C_{84}$

Order 312: $C_{13}:C_{24}$ x 2, $C_{52}:C_6$, $C_2\times F_{13}$, $C_{13}:C_{24}$, $C_{13}:C_{24}$

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

Order 224: $D_4:C_{28}$ x 2, $\OD_{16}:C_{14}$ x 2, $C_8.D_{14}$, $C_8.C_{28}$, $C_4\times C_{56}$, $D_8:C_{14}$

Order 208: $C_8\times D_{13}$, $C_{52}:C_4$, $D_{13}:C_8$

Order 196: $C_7\times C_{28}$ x 2, $C_7:C_{28}$ x 2, $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 192: $\GL(2,3):C_4$, $D_{24}:C_4$

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

Order 168: $C_{168}$ x 4, $C_2\times C_{84}$ x 2, $D_6:C_{14}$ x 2, $C_7\times D_{12}$ x 2, $S_3\times C_{28}$ x 2, $C_3:C_{56}$ x 2, $C_{21}:Q_8$ x 2, $C_7\times \SL(2,3)$

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

Order 112: $C_2\times C_{56}$ x 5, $C_7\times \OD_{16}$ x 3, $D_4:C_{14}$ x 2, $C_{56}:C_2$ x 2, $C_8\times D_7$ x 2, $C_7\times \SD_{16}$ x 2, $C_7:\OD_{16}$, $D_{28}:C_2$, $C_7\times Q_{16}$, $C_7\times D_8$, $C_4\times C_{28}$

Order 104: $C_{13}:C_8$ x 2, $C_4\times D_{13}$, $C_{104}$, $C_{26}:C_4$, $C_{13}:C_8$

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

Order 96: $D_{12}:C_4$ x 2, $C_8.D_6$ x 2, $C_8.A_4$, $\Unitary(2,3)$, $C_4\times C_{24}$, $C_{24}.C_4$, $\GL(2,3):C_2$, $D_{24}:C_2$

Order 91: $C_{91}$

Order 84: $C_{84}$ x 3, $C_3:C_{28}$ x 2, $S_3\times C_{14}$ x 2, $C_2\times C_{42}$

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

Order 64: $D_8:C_4$

Order 56: $C_{56}$ x 9, $C_2\times C_{28}$ x 5, $C_7\times D_4$ x 3, $C_7:D_4$ x 2, $C_4\times D_7$ x 2, $C_7\times Q_8$ x 2, $C_7:C_8$ x 2, $D_{28}$, $C_7:Q_8$

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

Order 49: $C_7^2$

Order 48: $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_{12}:C_2$ x 2, $C_2\times C_{24}$ x 2, $C_3:\OD_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_2.S_4$, $C_4\times C_{12}$

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

Order 39: $C_{13}:C_3$

Order 32: $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 28: $C_{28}$ x 8, $C_2\times C_{14}$ x 4, $D_{14}$ x 2, $C_7:C_4$ x 2

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

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 2, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $\SL(2,3)$

Order 21: $C_{21}$

Order 16: $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 14: $C_{14}$ x 7, $D_7$ x 2

Order 13: $C_{13}$

Order 12: $C_{12}$ x 3, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 7: $C_7$ x 3

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

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 122304: $C_{56}.\PGL(2,13)$ ($C_1$)

Order 61152: $\SL(2,13):C_{28}$ ($C_2$), $C_{28}.\PGL(2,13)$ ($C_2$), $C_{56}.\PSL(2,13)$ ($C_2$)

Order 30576: $C_{14}.\PGL(2,13)$ ($C_4$), $\SL(2,13):C_{14}$ ($C_2^2$), $\Unitary(2,13)$ ($C_4$)

Order 17472: $C_8.\PGL(2,13)$ ($C_7$)

Order 15288: $C_7\times \SL(2,13)$ ($C_2\times C_4$)

Order 8736: $C_4.\PGL(2,13)$ ($C_{14}$), $C_8.\PSL(2,13)$ ($C_{14}$), $\SL(2,13):C_4$ ($C_{14}$)

Order 4368: $\SL(2,13):C_2$ ($C_{28}$), $\SL(2,13):C_2$ ($C_2\times C_{14}$), $C_2.\PGL(2,13)$ ($C_{28}$)

Order 2184: $\SL(2,13)$ ($C_2\times C_{28}$)

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

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

Order 14: $C_{14}$ ($C_4.\PSL(2,13).C_2$)

Order 8: $C_8$ ($C_7.\PSL(2,13).C_2$)

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

Order 4: $C_4$ ($C_{14}.\PSL(2,13).C_2$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 122304: $C_{56}.\PGL(2,13)$ ($C_1$)

Order 61152: $\SL(2,13):C_{28}$ ($C_2$), $C_{28}.\PGL(2,13)$ ($C_2$), $C_{56}.\PSL(2,13)$ ($C_2$)

Order 30576: $C_{14}.\PGL(2,13)$ ($C_4$), $\SL(2,13):C_{14}$ ($C_2^2$), $\Unitary(2,13)$ ($C_4$)

Order 17472: $C_8.\PGL(2,13)$ ($C_7$)

Order 15288: $C_7\times \SL(2,13)$ ($C_2\times C_4$)

Order 8736: $C_4.\PGL(2,13)$ ($C_{14}$), $C_8.\PSL(2,13)$ ($C_{14}$), $\SL(2,13):C_4$ ($C_{14}$)

Order 4368: $\SL(2,13):C_2$ ($C_{28}$), $\SL(2,13):C_2$ ($C_2\times C_{14}$), $C_2.\PGL(2,13)$ ($C_{28}$)

Order 2184: $\SL(2,13)$ ($C_2\times C_{28}$)

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

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

Order 14: $C_{14}$ ($C_4.\PSL(2,13).C_2$)

Order 8: $C_8$ ($C_7.\PSL(2,13).C_2$)

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

Order 4: $C_4$ ($C_{14}.\PSL(2,13).C_2$)

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

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

Series

Derived series

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

$\rhd$

$\SL(2,13)$

Chief series

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

$\rhd$

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

$\rhd$

$C_{56}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$\SL(2,13)$

Upper central series

$C_1$

$\lhd$

$C_{56}$

Character theory

Complex character table

The $784 \times 784$ character table is not available for this group.

Rational character table

See the $89 \times 89$ rational character table.