Abstract group 188496.a: $C_{10472}.C_{18}$

• Label: 188496.a
• Order: \( 2^{4} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \cdot 17 \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \)
• Perm deg.: $52$
• Trans deg.: $94248$
• Rank: $2$

Group information

Description:	$C_{10472}.C_{18}$
Order:	\(188496\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
Exponent:	\(94248\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 17 \)
Automorphism group:	Group of order \(7096320\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_7$, $C_{11}$, $C_{17}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	6	7	8	9	11	12	14	17	18	21	22	24	28	33	34	36	42	44	51	56	63	66	68	72	77	84	88	99	102	119	126	132	136	153	154	168	187	198	204	231	238	252	264	306	308	357	374	396	408	462	476	504	561	612	616	693	714	748	792	924	952	1071	1122	1224	1309	1386	1428	1496	1683	1848	2142	2244	2618	2772	2856	3366	3927	4284	4488	5236	5544	6732	7854	8568	10472	11781	13464	15708	23562	31416	47124	94248
Elements	1	309	2	310	618	6	4	6	10	620	6	16	1854	12	10	8	12	20	4944	1860	12	20	32	24	36	20	4960	24	60	24	40	60	9888	96	36	40	64	96	60	48	160	60	9920	120	96	72	80	29664	120	192	160	120	128	120	192	144	320	29760	240	360	192	320	240	240	384	576	320	384	960	360	384	640	960	480	576	640	960	720	768	960	1920	1152	1280	1920	1440	1920	1920	2304	3840	5760	3840	3840	5760	7680	11520	23040	188496
Conjugacy classes	1	2	2	2	4	3	2	6	5	4	3	16	12	6	5	4	6	10	32	12	6	10	32	12	18	10	32	12	30	12	20	30	64	48	18	20	32	96	30	24	80	30	64	60	48	36	40	192	60	96	80	60	64	60	96	72	160	192	120	180	96	160	120	120	192	288	160	192	480	180	192	320	480	240	288	320	480	360	384	480	960	576	640	960	720	960	960	1152	1920	2880	1920	1920	2880	3840	5760	11520	47583
Divisions	1	2	1	2	2	1	1	1	1	2	1	1	2	1	1	1	1	1	2	2	1	1	1	1	1	1	2	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	108
Autjugacy classes	1	2	1	2	2	1	1	1	1	2	1	1	2	1	1	1	1	1	2	2	1	1	1	1	1	1	2	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	2	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	108

Minimal presentations

Permutation degree:	$52$
Transitive degree:	$94248$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Groups of Lie type:	$\COMinus(2,307)$
Presentation:	$\langle a, b \mid a^{2}=b^{94248}=1, b^{a}=b^{307} \rangle$
Permutation group:	Degree $52$ $\langle(1,2,3,4,6,8,5,7)(9,10,11,13,15,14,12)(16,17,18,19,20,21,22,23,24)(25,26,27,29,31,33,35,34,32,30,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 115 & 196 \\ 162 & 115 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 306 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{307})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{23562}$ . $D_4$	$C_{13464}$ . $D_7$	$C_{8568}$ . $D_{11}$	$C_{6732}$ . $D_{14}$	all 94

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

Homology

Abelianization:	$C_{2} \times C_{306} \simeq C_{2}^{2} \times C_{9} \times C_{17}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 6432 subgroups in 240 conjugacy classes, 114 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{306}$	$G/Z \simeq$ $D_{308}$
Commutator:	$G' \simeq$ $C_{308}$	$G/G' \simeq$ $C_2\times C_{306}$
Frattini:	$\Phi \simeq$ $C_{12}$	$G/\Phi \simeq$ $C_{102}\times D_{77}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{94248}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{10472}.C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{7854}$	$G/\operatorname{soc} \simeq$ $C_3\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 188496: $C_{10472}.C_{18}$

Order 94248: $C_{153}\times D_{308}$, $C_{94248}$, $C_{11781}:Q_8$

Order 62832: $C_{10472}.C_6$

Order 47124: $C_{47124}$, $C_{153}\times D_{154}$, $C_{77}:C_{612}$

Order 31416: $C_{3927}:Q_8$, $C_{31416}$, $C_{51}\times D_{308}$

Order 26928: $C_{88}:C_{306}$

Order 23562: $C_{23562}$, $C_{153}\times D_{77}$

Order 20944: $C_{10472}:C_2$

Order 17136: $C_{56}:C_{306}$

Order 15708: $C_{77}:C_{204}$, $C_{102}\times D_{77}$, $C_{15708}$

Order 13464: $C_{13464}$, $C_{1683}:Q_8$, $D_{44}\times C_{153}$

Order 11781: $C_{11781}$

Order 11088: $C_{5544}:C_2$

Order 10472: $C_{1309}:Q_8$, $C_{17}\times D_{308}$, $C_{10472}$

Order 8976: $C_{88}:C_{102}$

Order 8568: $C_{8568}$, $D_{28}\times C_{153}$, $C_{1071}:Q_8$

Order 7854: $C_{51}\times D_{77}$, $C_{7854}$

Order 6732: $C_{11}:C_{612}$, $C_{6732}$, $D_{11}\times C_{306}$

Order 5712: $C_{56}:C_{102}$

Order 5544: $C_9\times D_{308}$, $C_{693}:Q_8$, $C_{5544}$

Order 5236: $C_{17}\times C_{77}:C_4$, $C_{5236}$, $C_{17}\times D_{154}$

Order 4488: $C_{51}\times D_{44}$, $C_{561}:Q_8$, $C_{4488}$

Order 4284: $C_{306}\times D_7$, $C_{4284}$, $C_7:C_{612}$

Order 3927: $C_{3927}$

Order 3696: $C_{1848}:C_2$

Order 3366: $D_{11}\times C_{153}$, $C_{3366}$

Order 2992: $C_{17}\times C_8:D_{11}$

Order 2856: $C_{357}:Q_8$, $C_{51}\times D_{28}$, $C_{2856}$

Order 2772: $C_{18}\times D_{77}$, $C_{693}:C_4$, $C_{2772}$

Order 2618: $C_{17}\times D_{77}$, $C_{2618}$

Order 2448: $\SD_{16}\times C_{153}$

Order 2244: $C_{51}\times C_{11}:C_4$, $C_{102}\times D_{11}$, $C_{2244}$

Order 2142: $C_{153}\times D_7$, $C_{2142}$

Order 1904: $C_{56}:C_{34}$

Order 1848: $C_{231}:Q_8$, $C_{1848}$, $C_3\times D_{308}$

Order 1683: $C_{1683}$

Order 1584: $C_{792}:C_2$

Order 1496: $C_{1496}$, $C_{17}\times D_{44}$, $C_{187}:Q_8$

Order 1428: $C_7:C_{204}$, $D_7\times C_{102}$, $C_{1428}$

Order 1386: $C_9\times D_{77}$, $C_{1386}$

Order 1309: $C_{1309}$

Order 1232: $C_{616}:C_2$

Order 1224: $C_{1224}$, $Q_8\times C_{153}$, $D_4\times C_{153}$

Order 1122: $C_{1122}$, $D_{11}\times C_{51}$

Order 1071: $C_{1071}$

Order 1008: $C_{504}:C_2$

Order 952: $C_{952}$, $C_{17}\times D_{28}$, $C_{119}:Q_8$

Order 924: $C_{231}:C_4$, $C_3\times D_{154}$, $C_{924}$

Order 816: $\SD_{16}\times C_{51}$

Order 792: $C_{792}$, $C_9\times D_{44}$, $C_{99}:Q_8$

Order 748: $C_{17}\times D_{22}$, $C_{748}$, $C_{11}:C_{68}$

Order 714: $D_7\times C_{51}$, $C_{714}$

Order 693: $C_{693}$

Order 616: $C_{616}$, $D_{308}$, $C_{77}:Q_8$

Order 612: $C_{612}$ x 2, $C_2\times C_{306}$

Order 561: $C_{561}$

Order 528: $C_{264}:C_2$

Order 504: $C_{504}$, $C_9\times D_{28}$, $C_{63}:Q_8$

Order 476: $C_{17}\times D_{14}$, $C_{476}$, $C_7:C_{68}$

Order 462: $C_3\times D_{77}$, $C_{462}$

Order 408: $C_{408}$, $Q_8\times C_{51}$, $D_4\times C_{51}$

Order 396: $C_9\times D_{22}$, $C_{396}$, $C_{11}:C_{36}$

Order 374: $C_{374}$, $C_{17}\times D_{11}$

Order 357: $C_{357}$

Order 336: $C_{168}:C_2$

Order 308: $D_{154}$, $C_{308}$, $C_{77}:C_4$

Order 306: $C_{306}$ x 2

Order 272: $\SD_{16}\times C_{17}$

Order 264: $C_{264}$, $C_3\times D_{44}$, $C_{33}:Q_8$

Order 252: $C_{252}$, $C_7:C_{36}$, $C_9\times D_{14}$

Order 238: $C_{238}$, $D_7\times C_{17}$

Order 231: $C_{231}$

Order 204: $C_{204}$ x 2, $C_2\times C_{102}$

Order 198: $C_{198}$, $C_9\times D_{11}$

Order 187: $C_{187}$

Order 176: $C_{88}:C_2$

Order 168: $C_{168}$, $C_3\times D_{28}$, $C_{21}:Q_8$

Order 154: $C_{154}$, $D_{77}$

Order 153: $C_{153}$

Order 144: $C_9\times \SD_{16}$

Order 136: $C_{136}$, $Q_8\times C_{17}$, $D_4\times C_{17}$

Order 132: $C_3\times D_{22}$, $C_{132}$, $C_{11}:C_{12}$

Order 126: $C_{126}$, $C_9\times D_7$

Order 119: $C_{119}$

Order 112: $C_{56}:C_2$

Order 102: $C_{102}$ x 2

Order 99: $C_{99}$

Order 88: $D_{44}$, $C_{11}:Q_8$, $C_{88}$

Order 84: $C_{84}$, $C_7:C_{12}$, $C_3\times D_{14}$

Order 77: $C_{77}$

Order 72: $C_{72}$, $Q_8\times C_9$, $D_4\times C_9$

Order 68: $C_{68}$ x 2, $C_2\times C_{34}$

Order 66: $C_{66}$, $C_3\times D_{11}$

Order 63: $C_{63}$

Order 56: $D_{28}$, $C_7:Q_8$, $C_{56}$

Order 51: $C_{51}$

Order 48: $C_3\times \SD_{16}$

Order 44: $D_{22}$, $C_{44}$, $C_{11}:C_4$

Order 42: $C_{42}$, $C_3\times D_7$

Order 36: $C_{36}$ x 2, $C_2\times C_{18}$

Order 34: $C_{34}$ x 2

Order 33: $C_{33}$

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

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 22: $C_{22}$, $D_{11}$

Order 21: $C_{21}$

Order 18: $C_{18}$ x 2

Order 17: $C_{17}$

Order 16: $\SD_{16}$

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

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

Order 11: $C_{11}$

Order 9: $C_9$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$

Order 6: $C_6$ 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 188496: $C_{10472}.C_{18}$

Order 94248: $C_{94248}$

Order 62832: $C_{10472}.C_6$

Order 26928: $C_{88}:C_{306}$

Order 20944: $C_{10472}:C_2$

Order 17136: $C_{56}:C_{306}$

Order 11781: $C_{11781}$

Order 11088: $C_{5544}:C_2$

Order 8976: $C_{88}:C_{102}$

Order 5712: $C_{56}:C_{102}$

Order 3927: $C_{3927}$

Order 3696: $C_{1848}:C_2$

Order 2992: $C_{17}\times C_8:D_{11}$

Order 2448: $\SD_{16}\times C_{153}$

Order 1904: $C_{56}:C_{34}$

Order 1683: $C_{1683}$

Order 1584: $C_{792}:C_2$

Order 1309: $C_{1309}$

Order 1232: $C_{616}:C_2$

Order 1071: $C_{1071}$

Order 1008: $C_{504}:C_2$

Order 816: $\SD_{16}\times C_{51}$

Order 693: $C_{693}$

Order 561: $C_{561}$

Order 528: $C_{264}:C_2$

Order 357: $C_{357}$

Order 336: $C_{168}:C_2$

Order 272: $\SD_{16}\times C_{17}$

Order 231: $C_{231}$

Order 187: $C_{187}$

Order 176: $C_{88}:C_2$

Order 153: $C_{153}$

Order 144: $C_9\times \SD_{16}$

Order 119: $C_{119}$

Order 112: $C_{56}:C_2$

Order 99: $C_{99}$

Order 77: $C_{77}$

Order 63: $C_{63}$

Order 51: $C_{51}$

Order 48: $C_3\times \SD_{16}$

Order 33: $C_{33}$

Order 21: $C_{21}$

Order 17: $C_{17}$

Order 16: $\SD_{16}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 188496: $C_{10472}.C_{18}$ ($C_1$)

Order 94248: $C_{153}\times D_{308}$ ($C_2$), $C_{94248}$ ($C_2$), $C_{11781}:Q_8$ ($C_2$)

Order 62832: $C_{10472}.C_6$ ($C_3$)

Order 47124: $C_{47124}$ ($C_2^2$)

Order 31416: $C_{3927}:Q_8$ ($C_6$), $C_{31416}$ ($C_6$), $C_{51}\times D_{308}$ ($C_6$)

Order 23562: $C_{23562}$ ($D_4$)

Order 20944: $C_{10472}:C_2$ ($C_9$)

Order 15708: $C_{15708}$ ($C_2\times C_6$)

Order 13464: $C_{13464}$ ($D_7$)

Order 11781: $C_{11781}$ ($\SD_{16}$)

Order 11088: $C_{5544}:C_2$ ($C_{17}$)

Order 10472: $C_{1309}:Q_8$ ($C_{18}$), $C_{17}\times D_{308}$ ($C_{18}$), $C_{10472}$ ($C_{18}$)

Order 8568: $C_{8568}$ ($D_{11}$)

Order 7854: $C_{7854}$ ($C_3\times D_4$)

Order 6732: $C_{6732}$ ($D_{14}$)

Order 5544: $C_9\times D_{308}$ ($C_{34}$), $C_{693}:Q_8$ ($C_{34}$), $C_{5544}$ ($C_{34}$)

Order 5236: $C_{5236}$ ($C_2\times C_{18}$)

Order 4488: $C_{4488}$ ($C_3\times D_7$)

Order 4284: $C_{4284}$ ($D_{22}$)

Order 3927: $C_{3927}$ ($C_3\times \SD_{16}$)

Order 3696: $C_{1848}:C_2$ ($C_{51}$)

Order 3366: $C_{3366}$ ($D_{28}$)

Order 2856: $C_{2856}$ ($C_3\times D_{11}$)

Order 2772: $C_{2772}$ ($C_2\times C_{34}$)

Order 2618: $C_{2618}$ ($D_4\times C_9$)

Order 2244: $C_{2244}$ ($C_3\times D_{14}$)

Order 2142: $C_{2142}$ ($D_{44}$)

Order 1848: $C_{231}:Q_8$ ($C_{102}$), $C_{1848}$ ($C_{102}$), $C_3\times D_{308}$ ($C_{102}$)

Order 1683: $C_{1683}$ ($C_{56}:C_2$)

Order 1496: $C_{1496}$ ($C_9\times D_7$)

Order 1428: $C_{1428}$ ($C_3\times D_{22}$)

Order 1386: $C_{1386}$ ($D_4\times C_{17}$)

Order 1309: $C_{1309}$ ($C_9\times \SD_{16}$)

Order 1232: $C_{616}:C_2$ ($C_{153}$)

Order 1224: $C_{1224}$ ($D_{77}$)

Order 1122: $C_{1122}$ ($C_3\times D_{28}$)

Order 1071: $C_{1071}$ ($C_{88}:C_2$)

Order 952: $C_{952}$ ($C_9\times D_{11}$)

Order 924: $C_{924}$ ($C_2\times C_{102}$)

Order 792: $C_{792}$ ($D_7\times C_{17}$)

Order 748: $C_{748}$ ($C_9\times D_{14}$)

Order 714: $C_{714}$ ($C_3\times D_{44}$)

Order 693: $C_{693}$ ($\SD_{16}\times C_{17}$)

Order 616: $C_{616}$ ($C_{306}$), $D_{308}$ ($C_{306}$), $C_{77}:Q_8$ ($C_{306}$)

Order 612: $C_{612}$ ($D_{154}$)

Order 561: $C_{561}$ ($C_{168}:C_2$)

Order 504: $C_{504}$ ($C_{17}\times D_{11}$)

Order 476: $C_{476}$ ($C_9\times D_{22}$)

Order 462: $C_{462}$ ($D_4\times C_{51}$)

Order 408: $C_{408}$ ($C_3\times D_{77}$)

Order 396: $C_{396}$ ($C_{17}\times D_{14}$)

Order 374: $C_{374}$ ($C_9\times D_{28}$)

Order 357: $C_{357}$ ($C_{264}:C_2$)

Order 308: $C_{308}$ ($C_2\times C_{306}$)

Order 306: $C_{306}$ ($D_{308}$)

Order 264: $C_{264}$ ($D_7\times C_{51}$)

Order 252: $C_{252}$ ($C_{17}\times D_{22}$)

Order 238: $C_{238}$ ($C_9\times D_{44}$)

Order 231: $C_{231}$ ($\SD_{16}\times C_{51}$)

Order 204: $C_{204}$ ($C_3\times D_{154}$)

Order 198: $C_{198}$ ($C_{17}\times D_{28}$)

Order 187: $C_{187}$ ($C_{504}:C_2$)

Order 168: $C_{168}$ ($D_{11}\times C_{51}$)

Order 154: $C_{154}$ ($D_4\times C_{153}$)

Order 153: $C_{153}$ ($C_{616}:C_2$)

Order 136: $C_{136}$ ($C_9\times D_{77}$)

Order 132: $C_{132}$ ($D_7\times C_{102}$)

Order 126: $C_{126}$ ($C_{17}\times D_{44}$)

Order 119: $C_{119}$ ($C_{792}:C_2$)

Order 102: $C_{102}$ ($C_3\times D_{308}$)

Order 99: $C_{99}$ ($C_{56}:C_{34}$)

Order 88: $C_{88}$ ($D_7\times C_{153}$)

Order 84: $C_{84}$ ($D_{11}\times C_{102}$)

Order 77: $C_{77}$ ($\SD_{16}\times C_{153}$)

Order 72: $C_{72}$ ($C_{17}\times D_{77}$)

Order 68: $C_{68}$ ($C_9\times D_{154}$)

Order 66: $C_{66}$ ($C_{51}\times D_{28}$)

Order 63: $C_{63}$ ($C_{1496}.C_2$)

Order 56: $C_{56}$ ($D_{11}\times C_{153}$)

Order 51: $C_{51}$ ($C_{1848}:C_2$)

Order 44: $C_{44}$ ($D_7\times C_{306}$)

Order 42: $C_{42}$ ($C_{51}\times D_{44}$)

Order 36: $C_{36}$ ($C_{17}\times D_{154}$)

Order 34: $C_{34}$ ($C_9\times D_{308}$)

Order 33: $C_{33}$ ($C_{56}:C_{102}$)

Order 28: $C_{28}$ ($D_{11}\times C_{306}$)

Order 24: $C_{24}$ ($C_{51}\times D_{77}$)

Order 22: $C_{22}$ ($D_{28}\times C_{153}$)

Order 21: $C_{21}$ ($C_{88}:C_{102}$)

Order 18: $C_{18}$ ($C_{17}\times D_{308}$)

Order 17: $C_{17}$ ($C_{5544}:C_2$)

Order 14: $C_{14}$ ($D_{44}\times C_{153}$)

Order 12: $C_{12}$ ($C_{102}\times D_{77}$)

Order 11: $C_{11}$ ($C_{56}:C_{306}$)

Order 9: $C_9$ ($C_{10472}:C_2$)

Order 8: $C_8$ ($C_{153}\times D_{77}$)

Order 7: $C_7$ ($C_{88}:C_{306}$)

Order 6: $C_6$ ($C_{51}\times D_{308}$)

Order 4: $C_4$ ($C_{153}\times D_{154}$)

Order 3: $C_3$ ($C_{10472}.C_6$)

Order 2: $C_2$ ($C_{153}\times D_{308}$)

Order 1: $C_1$ ($C_{10472}.C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 188496: $C_{10472}.C_{18}$ ($C_1$)

Order 94248: $C_{94248}$ ($C_2$)

Order 62832: $C_{10472}.C_6$ ($C_3$)

Order 20944: $C_{10472}:C_2$ ($C_9$)

Order 11781: $C_{11781}$ ($\SD_{16}$)

Order 11088: $C_{5544}:C_2$ ($C_{17}$)

Order 3927: $C_{3927}$ ($C_3\times \SD_{16}$)

Order 3696: $C_{1848}:C_2$ ($C_{51}$)

Order 1683: $C_{1683}$ ($C_{56}:C_2$)

Order 1309: $C_{1309}$ ($C_9\times \SD_{16}$)

Order 1232: $C_{616}:C_2$ ($C_{153}$)

Order 1071: $C_{1071}$ ($C_{88}:C_2$)

Order 693: $C_{693}$ ($\SD_{16}\times C_{17}$)

Order 561: $C_{561}$ ($C_{168}:C_2$)

Order 357: $C_{357}$ ($C_{264}:C_2$)

Order 231: $C_{231}$ ($\SD_{16}\times C_{51}$)

Order 187: $C_{187}$ ($C_{504}:C_2$)

Order 153: $C_{153}$ ($C_{616}:C_2$)

Order 119: $C_{119}$ ($C_{792}:C_2$)

Order 99: $C_{99}$ ($C_{56}:C_{34}$)

Order 77: $C_{77}$ ($\SD_{16}\times C_{153}$)

Order 63: $C_{63}$ ($C_{1496}.C_2$)

Order 51: $C_{51}$ ($C_{1848}:C_2$)

Order 33: $C_{33}$ ($C_{56}:C_{102}$)

Order 21: $C_{21}$ ($C_{88}:C_{102}$)

Order 17: $C_{17}$ ($C_{5544}:C_2$)

Order 11: $C_{11}$ ($C_{56}:C_{306}$)

Order 9: $C_9$ ($C_{10472}:C_2$)

Order 7: $C_7$ ($C_{88}:C_{306}$)

Order 3: $C_3$ ($C_{10472}.C_6$)

Order 1: $C_1$ ($C_{10472}.C_{18}$)

Series

Derived series

$C_{10472}.C_{18}$

$\rhd$

$C_{308}$

$\rhd$

$C_1$

Chief series

$C_{10472}.C_{18}$

$\rhd$

$C_{153}\times D_{308}$

$\rhd$

$C_{47124}$

$\rhd$

$C_{23562}$

$\rhd$

$C_{11781}$

$\rhd$

$C_{3927}$

$\rhd$

$C_{1309}$

$\rhd$

$C_{187}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{10472}.C_{18}$

$\rhd$

$C_{308}$

$\rhd$

$C_{154}$

$\rhd$

$C_{77}$

Upper central series

$C_1$

$\lhd$

$C_{306}$

$\lhd$

$C_{612}$

$\lhd$

$C_{1224}$

Character theory

Complex character table

The $47583 \times 47583$ character table is not available for this group.

Rational character table

The $108 \times 108$ rational character table is not available for this group.