Abstract group 685597979049984000.a: $C_2^{15}.A_{16}.C_2$

• Label: 685...000.a
• Order: \( 2^{30} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{31} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$C_2^{15}.A_{16}.C_2$
Order:	\(685597979049984000\)\(\medspace = 2^{30} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
Exponent:	\(720720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \)
Automorphism group:	Group of order \(1371195958099968000\)\(\medspace = 2^{31} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \)
Composition factors:	$C_2$ x 16, $A_{16}$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	18	20	21	22	24	26	28	30	33	35	36	39	40	42	44	45	48	52	55	56	60	63	66	70	72	78	80	84	88	90	105	110	112	120	126	132	140	168	180	210	240	280
Elements	1	25412227327	851185059200	234994584334080	114415115802624	2672259121347200	437244136243200	10370763648860160	562192043212800	9195147295746048	16231012761600	48346246268436480	1098714710016000	29384406293299200	22986016826572800	52142514949324800	15812581117132800	25910505502801920	104267081318400	5047844968857600	57777503207424000	20875579490304000	30270466149580800	44090543469772800	1298481020928000	25505877196800	24162567664435200	8789717680128000	18928761663651840	9869475564748800	16231012761600000	952219415347200	16187730060902400	13184576520192000	6232708900454400	11069550703411200	37673999094251520	5441253801984000	14283291230208000	3851387456716800	9522194153472000	8789717680128000	5713316492083200	18109172809728000	7790886125568000	10474413568819200	816188070297600	6232708900454400	6121410527232000	10117331288064000	5441253801984000	5193924083712000	6733551579955200	4080940351488000	3808877661388800	5713316492083200	2856658246041600	2448564210892800	685597979049984000
Conjugacy classes	1	45	5	214	3	248	2	195	3	82	1	663	1	39	6	44	43	174	3	11	300	5	65	153	1	1	42	1	76	59	10	1	36	2	1	26	187	1	7	17	10	1	8	56	2	7	1	1	2	44	1	2	12	10	2	7	2	2	2944
Divisions	1	45	5	214	3	248	2	195	3	82	1	663	1	39	6	44	43	174	3	11	300	5	65	153	1	1	42	1	76	59	10	1	36	2	1	26	187	1	7	17	10	1	8	56	2	7	1	1	2	44	1	2	12	10	2	7	2	2	2944
Autjugacy classes	1	34	5	170	3	190	2	157	3	63	1	509	1	30	6	36	35	132	3	9	231	4	49	120	1	1	29	1	58	47	7	1	28	1	1	19	136	1	6	14	7	1	6	39	1	6	1	1	1	32	1	1	8	7	1	7	1	1	2267

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $32$ $\langle(1,28,9,24,31,12,19,4,6,26,14,17,15,22,30,8)(2,27,10,23,32,11,20,3,5,25,13,18,16,21,29,7) \!\cdots\! \rangle$
Transitive group:	32T2801272				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{15}$ . $S_{16}$	$(C_2^{15}.A_{16})$ . $C_2$	$C_2$ . $(C_2^{14}.A_{16}.C_2)$		more information

Elements of the group are displayed as permutations of degree 32.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 5 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^{15}.A_{16}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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

All subgroups of index up to 130977 (order at least 5234491392000) are shown, as well as all normal subgroups of any index.

Order 685597979049984000: $C_2^{15}.A_{16}.C_2$

Order 342798989524992000: $C_2^{15}.A_{16}$

Order 42849873690624000: $C_2^{15}.A_{15}.C_2$

Order 21424936845312000: $C_2^{14}.A_{15}.C_2$ x 2, $C_2^{15}.A_{15}$

Order 10712468422656000: $C_2^{14}.A_{15}$

Order 5713316492083200: $C_2^{15}.C_2.A_{14}.C_2$

Order 2856658246041600: $C_2^{15}.A_{14}.C_2$ x 4, $(C_2^{13}\times C_4).A_{14}.C_2$ x 2, $C_2^{15}.C_2.A_{14}$

Order 1428329123020800: $C_2^{14}.A_{14}.C_2$ x 6, $C_2^{15}.A_{14}$ x 2, $C_2^{14}.A_{14}.C_2$ x 2, $(C_2^{13}\times C_4).A_{14}$

Order 1224282105446400: $(C_2\times S_4).C_2^{12}.A_{13}.C_2$

Order 714164561510400: $C_2^{13}.A_{14}.C_2$ x 3, $C_2^{14}.A_{14}$ x 2, $C_2^{13}.A_{14}.C_2$, $C_2^{14}.A_{14}$

Order 612141052723200: $A_4.C_2^{12}.C_2.A_{13}.C_2$ x 5, $S_4.C_2^{12}.A_{13}.C_2$, $A_4.C_2^{12}.C_2^2.A_{13}$

Order 408094035148800: $C_2^{14}.C_2^2.A_{13}.C_2$

Order 376702186291200: $(C_2\times C_2^3:A_4).C_2^{10}.C_2^2.A_{12}.C_2$

Order 357082280755200: $C_2^{13}.A_{14}$

Order 306070526361600: $A_4.C_2^{12}.A_{13}.C_2$ x 4, $A_4.C_2^{12}.C_2.A_{13}$ x 2, $S_4.C_2^{12}.A_{13}$, $S_3.C_2^{12}.C_2.A_{13}.C_2$

Order 204047017574400: $C_2^{15}.A_{13}.C_2$ x 4, $C_2^{14}.C_2.A_{13}.C_2$ x 4, $D_4.C_2^{12}.A_{13}.C_2$ x 2, $(C_2^{13}\times C_4).A_{13}.C_2$ x 2, $C_2^{14}.C_2^2.A_{13}$, $C_2^{13}.C_2^2.A_{13}.C_2$, $(C_2^{12}\times C_4).C_2.A_{13}.C_2$

Order 188351093145600: $(C_2\times C_2^3:A_4).C_2^{10}.C_2.A_{12}.C_2$ x 5, $C_2^3:S_4.C_2.C_2^{10}.C_2.A_{12}$, $C_2^3:S_4.C_2.C_2^{10}.A_{12}.C_2$

Order 156959244288000: $C_2^{15}.A_{11}.A_5.C_2^2$

Order 153035263180800: $(C_2^{11}\times C_6).C_2.A_{13}.C_2$ x 3, $(C_2^{12}\times C_6).A_{13}.C_2$ x 2, $S_3.C_2^{12}.C_2.A_{13}$, $S_3.C_2^{12}.A_{13}.C_2$, $A_4.C_2^{12}.A_{13}$

Order 125567395430400: $(C_2^2\times D_4).C_2^{10}.C_2^3.A_{12}.C_2$

Order 106542032486400: $C_2^{15}.A_8^2.D_4$

Order 102023508787200: $C_2^{14}.A_{13}.C_2$ x 24, $(C_2^{12}\times C_4).A_{13}.C_2$ x 8, $C_2^{15}.A_{13}$ x 2, $C_2^{14}.C_2.A_{13}$ x 2, $D_4.C_2^{12}.A_{13}$, $(C_2^{13}\times C_4).A_{13}$, $(C_2^{12}\times C_4).C_2.A_{13}$

Order 94175546572800: $(C_2\times C_2^3:A_4).C_2^{10}.A_{12}.C_2$ x 4, $(C_2\times C_2^3:A_4).C_2^{10}.C_2.A_{12}$ x 3, $(C_2\times A_4).C_2^{10}.C_2^3.A_{12}.C_2$

Order 85614133248000: $C_2^{15}.A_{10}.A_6.C_2^2$

Order 78479622144000: $C_2^{15}.A_{11}.S_5$ x 2, $C_2^{14}.A_{11}.S_5.C_2$ x 2, $C_2^{14}.A_{11}.A_5.C_2^2$ x 2, $C_2^{15}.A_{11}.A_5.C_2$

Order 76517631590400: $(C_2^{11}\times C_6).A_{13}.C_2$ x 4, $S_3.C_2^{12}.A_{13}$, $(C_2^{12}\times C_6).A_{13}$, $(C_2^{11}\times C_6).C_2.A_{13}$

Order 62783697715200: $(C_2^2\times D_4).C_2^{10}.C_2^2.A_{12}.C_2$ x 5, $(C_2\times C_2^2.D_4).C_2^{10}.C_2.A_{12}.C_2$ x 3, $C_2^{14}.C_2^3.A_{12}.C_2$ x 2, $(C_2\times Q_8:C_2^2).C_2^{10}.C_2.A_{12}.C_2$ x 2, $C_2^{15}.C_2^2.A_{12}.C_2$, $C_2^4.C_2^3.C_2^{10}.A_{12}.C_2$, $(C_2^2\times D_4).C_2^{10}.C_2^3.A_{12}$

Order 59929893273600: $C_2^{15}.A_9.A_7.C_2^2$

Order 53271016243200: $C_2^{14}.A_8^2.D_4$ x 2, $C_2^{14}.A_8^2.D_4$ x 2, $C_2^{15}.A_8.A_8.C_2^2$, $C_2^{15}.A_8^2.C_2^2$, $C_2^{15}.A_8^2.C_4$

Order 51011754393600: $C_2^{12}.(C_2\times S_{13})$ x 21, $C_2^{14}.A_{13}$ x 7, $C_2^{13}.A_{13}.C_2$ x 2, $(C_2^{12}\times C_4).A_{13}$ x 2

Order 47087773286400: $(C_2\times A_4).C_2^{10}.C_2^2.A_{12}.C_2$ x 9, $(C_2\times S_4).C_2^{10}.C_2.A_{12}.C_2$ x 3, $(C_2^2\times A_4).C_2^{10}.C_2.A_{12}.C_2$ x 2, $(C_2^2\times S_4).C_2^{10}.A_{12}.C_2$, $(C_2\times \SL(2,3)).C_2^{10}.C_2.A_{12}.C_2$, $(C_2\times C_2^3:A_4).C_2^{10}.A_{12}$, $(C_2\times A_4).C_2^{10}.C_2^3.A_{12}$

Order 42807066624000: $C_2^{14}.A_{10}.A_6.C_2^2$ x 4, $C_2^{15}.A_{10}.A_6.C_2$ x 3

Order 41845579776000: $C_2.S_{16}$ x 2

Order 39239811072000: $C_2^{14}.A_{11}.S_5$ x 4, $C_2^{14}.A_{11}.A_5.C_2$ x 2, $C_2^{15}.A_{11}.A_5$

Order 38258815795200: $(C_2^{11}\times C_6).A_{13}$

Order 31391848857600: $C_2^{14}.C_2^2.A_{12}.C_2$ x 20, $(C_2^2\times D_4).C_2^{10}.C_2.A_{12}.C_2$ x 7, $C_2^{13}.C_2^3.A_{12}.C_2$ x 6, $(C_2^{12}\times C_4).C_2^2.A_{12}.C_2$ x 5, $(C_2\times C_2^2.D_4).C_2^{10}.A_{12}.C_2$ x 4, $C_2^{15}.C_2.A_{12}.C_2$ x 2, $C_2^{14}.C_2^3.A_{12}$ x 2, $(C_2\times C_2^2:C_4).C_2^{10}.C_2.A_{12}.C_2$ x 2, $C_2^3:A_4.C_2^{10}.C_2^3.A_{11}.C_2$, $(C_2^{12}\times C_4).C_2^3.A_{12}$, $(C_2^{11}\times C_4).C_2^3.A_{12}.C_2$, $(C_2^2\times D_4).C_2^{10}.C_2^2.A_{12}$, $(C_2\times Q_8:C_2^2).C_2^{10}.A_{12}.C_2$, $(C_2\times C_2^2\wr C_2).C_2^{10}.C_2.A_{12}$, $(C_2\times C_2^2:C_4).C_2^{10}.C_2^2.A_{12}$, $(C_2\times C_2^2.D_4).C_2^{10}.C_2.A_{12}$

Order 29964946636800: $C_2^{14}.A_9.A_7.C_2^2$ x 4, $C_2^{15}.A_9.A_7.C_2$ x 3

Order 26635508121600: $C_2^{14}.A_8.A_8.C_2^2$ x 3, $C_2^{15}.A_8.A_8.C_2$ x 2, $C_2^{14}.A_8^2.C_2^2$ x 2, $C_2^{14}.A_8^2.C_4$ x 2, $C_2^{15}.A_8^2.C_2$, $C_2^{14}.A_8^2.C_2^2$

Order 26159874048000: $C_2^4:D_5.C_2^{10}.C_2^2.A_{11}.C_2$

Order 25505877196800: $C_2^{12}.S_{13}$ x 6, $C_2^{13}.A_{13}$ x 5

Order 23543886643200: $(C_2\times A_4).C_2^{10}.C_2.A_{12}.C_2$ x 28, $(C_2\times A_4).C_2^{10}.C_2^2.A_{12}$ x 6, $(C_2^2\times A_4).C_2^{10}.A_{12}.C_2$ x 3, $(C_2\times S_4).C_2^{10}.A_{12}.C_2$ x 3, $(C_2\times \SL(2,3)).C_2^{10}.A_{12}.C_2$ x 2, $(C_2\times S_4).C_2^{10}.C_2.A_{12}$ x 2, $(C_2^{12}\times C_6).C_2.A_{12}.C_2$, $(C_2\times \SL(2,3)).C_2^{10}.C_2.A_{12}$

Order 21403533312000: $C_2^{14}.A_{10}.A_6.C_2$ x 6, $C_2^{15}.A_{10}.A_6$

Order 20922789888000: $S_{16}$ x 4, $C_2.A_{16}$ x 2

Order 19619905536000: $C_2^{14}.A_{11}.A_5$

Order 15695924428800: $C_2^{14}.C_2.A_{12}.C_2$ x 43, $C_2^{13}.C_2^2.A_{12}.C_2$ x 30, $C_2^{15}.A_{12}.C_2$ x 11, $(C_2^{12}\times C_4).C_2.A_{12}.C_2$ x 10, $C_2^3:A_4.C_2^{10}.C_2^2.A_{11}.C_2$ x 9, $(C_2^{13}\times C_4).A_{12}.C_2$ x 8, $C_2^{14}.C_2^2.A_{12}$ x 6, $C_2^{12}.C_2^3.A_{12}.C_2$ x 6, $C_2^{13}.C_2^3.A_{12}$ x 4, $(C_2^{11}\times C_4^2).A_{12}.C_2$ x 4, $(C_2^{11}\times C_4).C_2^2.A_{12}.C_2$ x 4, $(C_2^{12}\times C_4).C_2^2.A_{12}$ x 3, $(C_2^2\times D_4).C_2^{10}.A_{12}.C_2$ x 3, $(C_2\times C_2^2:C_4).C_2^{10}.A_{12}.C_2$ x 3, $C_2^3:S_4.C_2^{10}.C_2.A_{11}.C_2$ x 2, $(C_2^{11}\times C_8).C_2.A_{12}.C_2$ x 2, $(C_2^2\times D_4).C_2^{10}.C_2.A_{12}$ x 2, $C_2^3:A_4:C_2.C_2^{10}.C_2.A_{11}.C_2$, $C_2^3:A_4.C_2^{10}.C_2^3.A_{11}$, $C_2\wr A_4.C_2^{10}.C_2.A_{11}.C_2$, $C_2\wr A_4.C_2.C_2^{10}.A_{11}.C_2$, $(C_2^{11}\times C_4^2).C_2.A_{12}$, $(C_2^2\times A_4).C_2^{10}.C_2^3.A_{11}.C_2$, $(C_2\times Q_8).C_2^{10}.C_2.A_{12}.C_2$, $(C_2\times D_4:C_2).C_2^{10}.C_2.A_{12}$, $(C_2\times D_4:C_2).C_2^{10}.A_{12}.C_2$, $(C_2\times D_4).C_2^{10}.C_2.A_{12}.C_2$

Order 14982473318400: $C_2^{14}.A_9.A_7.C_2$ x 6, $C_2^{15}.A_9.A_7$

Order 14269022208000: $C_2^{15}.A_{10}.S_5.C_2$, $C_2^{15}.A_{10}.A_5.C_2^2$

Order 13317754060800: $C_2^{14}.A_8.A_8.C_2$ x 4, $C_2^{14}.A_8^2.C_2$ x 2, $C_2^{15}.A_8.A_8$

Order 13079937024000: $C_2^4:D_5.C_2^{10}.C_2.A_{11}.C_2$ x 3, $C_2^4:C_5.C_2^{10}.C_2^2.A_{11}.C_2$ x 2, $C_2^4:D_5.C_2^2.C_2^{10}.A_{11}$, $C_2^4:C_5:C_4.C_2^{10}.A_{11}.C_2$

Order 12752938598400: $C_2^{12}.A_{13}$

Order 11771943321600: $(C_2\times A_4).C_2^{10}.A_{12}.C_2$ x 12, $(C_2\times A_4).C_2^{10}.C_2.A_{12}$ x 7, $(C_2^{10}\times C_6).C_2^2.A_{12}.C_2$ x 5, $D_6.C_2^{10}.C_2.A_{12}.C_2$ x 4, $(C_2^{12}\times C_6).A_{12}.C_2$ x 2, $(C_2^{11}\times C_6).C_2.A_{12}.C_2$ x 2, $(C_2^2\times A_4).C_2^{10}.A_{12}$ x 2, $D_6.C_2^{10}.C_2^2.A_{12}$, $(C_2^2\times S_3).C_2^{10}.A_{12}.C_2$, $(C_2\times \SL(2,3)).C_2^{10}.A_{12}$, $(C_2\times S_4).C_2^{10}.A_{12}$

Order 10701766656000: $C_2^{14}.A_{10}.A_6$

Order 10463949619200: $D_4^2.C_2^2.C_2^{10}.A_{11}.C_2$

Order 10461394944000: $A_{16}$ x 2

Order 9809952768000: $C_2^{11}.A_{11}.S_5.C_2$

Order 8561413324800: $C_2^{15}.A_9.A_6.C_2^2$, $A_4^2.C_2^3.C_2^8.C_2^3.A_{10}.C_2$

Order 7847962214400: $C_2^{14}.A_{12}.C_2$ x 79, $C_2^{13}.C_2.A_{12}.C_2$ x 57, $(C_2^{12}\times C_4).A_{12}.C_2$ x 56, $C_2^3:A_4.C_2^{10}.C_2.A_{11}.C_2$ x 19, $(C_2\times A_4).C_2^{10}.C_2^3.A_{11}.C_2$ x 12, $C_2^{13}.C_2^2.A_{12}$ x 10, $C_2^{12}.C_2^2.A_{12}.C_2$ x 10, $(C_2^{11}\times C_4).C_2.A_{12}.C_2$ x 9, $C_2^{14}.C_2.A_{12}$ x 8, $(C_2\times D_4).C_2^{10}.A_{12}.C_2$ x 7, $(C_2\times S_4).C_2^{10}.C_2^2.A_{11}.C_2$ x 5, $C_2^3:A_4:C_2.C_2^{10}.A_{11}.C_2$ x 4, $(C_2^{11}\times C_8).A_{12}.C_2$ x 4, $(C_2^2\times A_4).C_2^{10}.C_2^2.A_{11}.C_2$ x 4, $C_2^{15}.A_{12}$ x 3, $C_2^3:S_4.C_2^{10}.C_2.A_{11}$ x 3, $C_2^3:S_4.C_2^{10}.A_{11}.C_2$ x 3, $(C_2^{12}\times C_4).C_2.A_{12}$ x 3, $(C_2\times D_4).C_2^{10}.C_2.A_{12}$ x 3, $C_2^3:A_4.C_2^{10}.C_2^2.A_{11}$ x 2, $C_2\wr A_4.C_2^{10}.A_{11}.C_2$ x 2, $A_4.C_2^{10}.C_2^4.A_{11}.C_2$ x 2, $(D_4\times A_4).C_2^{10}.C_2.A_{11}.C_2$ x 2, $(C_2^{13}\times C_4).A_{12}$ x 2, $(C_2\times C_2^2:C_4).C_2^{10}.A_{12}$ x 2, $\GL(2,\mathbb{Z}/4).C_2^{10}.C_2.A_{11}.C_2$, $C_4:S_4.C_2^{10}.C_2.A_{11}.C_2$, $C_2^{12}.C_2^3.A_{12}$, $C_2^3:A_4:C_2.C_2^{10}.C_2.A_{11}$, $C_2\wr A_4.C_2^{10}.C_2.A_{11}$, $(D_4\times S_4).C_2^{10}.C_2.A_{11}$, $(C_4\times S_4).C_2^{10}.C_2.A_{11}.C_2$, $(C_4\times A_4).C_2^{10}.C_2^2.A_{11}.C_2$, $(C_2^{11}\times C_8).C_2.A_{12}$, $(C_2^{11}\times C_4^2).A_{12}$, $(C_2^{11}\times C_4).C_2^2.A_{12}$, $(C_2^2\times D_4).C_2^{10}.A_{12}$, $(C_2\times \GL(2,\mathbb{Z}/4)).C_2^{10}.A_{11}.C_2$, $(C_2\times Q_8).C_2^{10}.C_2.A_{12}$

Order 7491236659200: $C_2^{14}.A_9.A_7$

Order 7134511104000: $C_2^{14}.A_{10}.S_5.C_2$ x 13, $C_2^{15}.A_{10}.S_5$ x 4, $C_2^{14}.A_{10}.A_5.C_2^2$ x 3, $C_2^{15}.A_{10}.A_5.C_2$ x 2

Order 6658877030400: $C_2^{15}.A_8.A_7.C_2^2$, $C_2^{14}.A_8.A_8$

Order 6539968512000: $C_2^4:D_5.C_2^{10}.A_{11}.C_2$ x 4, $C_2^4:C_5.C_2^{10}.C_2.A_{11}.C_2$ x 4, $C_2^4:D_5.C_2^{10}.C_2.A_{11}$ x 2, $C_2^4:C_5:C_4.C_2^{10}.A_{11}$

Order 5885971660800: $(C_2^{11}\times C_6).A_{12}.C_2$ x 12, $(C_2^{10}\times C_6).C_2.A_{12}.C_2$ x 12, $D_6.C_2^{10}.A_{12}.C_2$ x 4, $D_6.C_2^{10}.C_2.A_{12}$ x 3, $(C_2^{10}\times C_6).C_2^2.A_{12}$ x 2, $(C_2\times A_4).C_2^{10}.A_{12}$ x 2, $(C_2^{12}\times C_6).A_{12}$, $(C_2^{11}\times C_6).C_2.A_{12}$

Order 5707608883200: $C_2^5.S_4.C_2.C_2^8.C_2^2.A_{10}.C_2$, $C_2^3:A_4.C_2^3.C_2^8.C_2^3.A_{10}.C_2$

Order 32768: $C_2^{15}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 685597979049984000: $C_2^{15}.A_{16}.C_2$ ($C_1$)

Order 342798989524992000: $C_2^{15}.A_{16}$ ($C_2$)

Order 32768: $C_2^{15}$ ($S_{16}$)

Order 2: $C_2$ ($C_2^{14}.A_{16}.C_2$)

Order 1: $C_1$ ($C_2^{15}.A_{16}.C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $2944 \times 2944$ rational character table is not available for this group.