Abstract group 839171926357180416000.a: $C_2^{18}.A_{18}$

• Label: 839...000.a
• Order: \( 2^{33} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{34} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^{18}.A_{18}$
Order:	\(839\!\cdots\!000\)\(\medspace = 2^{33} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \)
Exponent:	\(24504480\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \)
Automorphism group:	Group of order \(167\!\cdots\!000\)\(\medspace = 2^{34} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \)
Composition factors:	$C_2$ x 18, $A_{18}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	20	21	22	24	26	28	30	32	33	34	35	36	39	40	42	44	45	48	52	55	56	60	63	65	66	70	72	77	78	80	84	88	90	105	110	112	120	126	130	132	140	154	168	180	210	240	280
Elements	1	1183874212863	88611696649088	53495631152225280	5828395825050624	1119890379486026880	11149713498193920	5421283663050178560	2689299573011251200	4752002430927424512	118254521548800	41163807788024463360	16810335063244800	8738221152924057600	10554241743268134912	34575257010438144000	24681527245799424000	15570749002294886400	19848420807542046720	368788984971264000	824825287802880000	61697027883191500800	5093531524163174400	40232063409743462400	56939413782328197120	26224122698661888000	562891522572288000	74044581737398272000	299964419360686080	22454405060729241600	1344826805059584000	24013803785488957440	9223721228235571200	11920055772119040000	801292638014668800	13986198772619673600	12103441245536256000	953604461769523200	24163655915195596800	42202019308649840640	832511831703552000	6455168664286003200	11622054377816064000	5311165326321254400	10198269938368512000	5449168352968704000	20172402075893760000	17482748465774592000	17365676489441280000	9536044617695232000	14350422698989977600	208127957925888000	14304066926542848000	7492606485331968000	17045679754130227200	12487677475553280000	19365505992858009600	4768022308847616000	5244824539732377600	16347505058906112000	9365758106664960000	3496549693154918400	5119947764976844800	3496549693154918400	2997042594132787200	839171926357180416000
Conjugacy classes	1	54	6	304	3	403	2	333	5	136	1	1254	1	67	10	90	2	111	333	4	23	697	15	129	363	2	2	6	2	104	1	178	156	24	3	104	8	1	66	475	1	2	38	54	32	2	15	26	152	8	53	2	15	8	144	15	6	8	32	6	40	8	46	8	8	6208
Divisions	1	54	6	304	3	403	2	333	5	136	1	1254	1	67	9	90	1	111	333	4	23	697	15	129	360	2	2	3	2	104	1	178	152	24	2	104	8	1	66	475	1	1	34	50	32	1	11	26	152	8	38	2	11	8	144	11	3	8	32	3	40	8	42	8	8	6148
Autjugacy classes	1	44	6	244	3	312	2	272	5	105	1	967	1	52	9	75	1	82	256	4	17	542	11	98	270	2	2	2	2	78	1	137	113	18	2	81	6	1	50	357	1	1	25	37	24	1	8	20	114	6	28	2	8	6	108	8	2	6	24	2	30	6	31	6	6	4742

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $36$ $\langle(1,22,7,27,33,32,23,35,9,25,5,3,13,11,18,2,21,8,28,34,31,24,36,10,26,6,4,14,12,17) \!\cdots\! \rangle$
Transitive group:	36T121215				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{18}$ . $A_{18}$	$(C_2^{17}.A_{18})$ . $C_2$	$C_2^{17}$ . $(C_2.A_{18})$	$C_2$ . $(C_2^{17}.A_{18})$	more information

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 6 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 $C_2^{17}.A_{18}$
Frattini:	not computed
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^6.C_2^6.C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
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

All subgroups of index up to 219912 (order at least 3815944224768000) are shown, as well as all normal subgroups of any index.

Order 839171926357180416000: $C_2^{18}.A_{18}$

Order 419585963178590208000: $C_2^{17}.A_{18}$

Order 46620662575398912000: $C_2^{18}.A_{17}$

Order 23310331287699456000: $C_2^{17}.A_{17}$ x 3

Order 11655165643849728000: $C_2^{16}.A_{17}$

Order 5484783832399872000: $C_2^{18}.A_{16}.C_2$

Order 2742391916199936000: $C_2^{17}.A_{16}.C_2$ x 6, $C_2^{18}.A_{16}$

Order 1371195958099968000: $C_2^{17}.A_{16}$ x 5, $C_2^{16}.A_{16}.C_2$ x 4, $C_2^{16}.A_{16}.C_2$ x 2

Order 1028396968574976000: $(C_2\times A_4).C_2^{14}.C_2.A_{15}.C_2$

Order 685597979049984000: $C_2^{16}.A_{16}$ x 3, $C_2^{16}.A_{16}$ x 2, $C_2^{15}.A_{16}.C_2$, $C_2^{15}.A_{16}.C_2$

Order 514198484287488000: $A_4.C_2^{14}.C_2.A_{15}.C_2$ x 4, $(C_2\times A_4).C_2^{14}.A_{15}.C_2$ x 2, $A_4.C_2^{14}.C_2^2.A_{15}$

Order 342798989524992000: $C_2^{18}.A_{15}.C_2$, $C_2^{15}.A_{16}$

Order 274239191619993600: $C_2\wr A_4.C_2.C_2^{12}.C_2.A_{14}.C_2$

Order 257099242143744000: $A_4.C_2^{14}.A_{15}.C_2$ x 4, $A_4.C_2^{14}.C_2.A_{15}$ x 2, $(C_2^{15}\times C_6).A_{15}.C_2$, $(C_2\times A_4).C_2^{14}.A_{15}$

Order 171399494762496000: $C_2^{17}.A_{15}.C_2$ x 14, $C_2^{18}.A_{15}$

Order 137119595809996800: $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{14}.C_2$ x 5, $C_2\wr A_4.C_2.C_2^{12}.A_{14}.C_2$, $(C_2\times C_2^3:A_4).C_2^{12}.C_2^2.A_{14}$

Order 128549621071872000: $(C_2^{14}\times C_6).A_{15}.C_2$ x 6, $A_4.C_2^{14}.A_{15}$, $(C_2^{15}\times C_6).A_{15}$

Order 97942568435712000: $C_2^{18}.A_{13}.S_5$

Order 91413063873331200: $C_2^{17}.C_2^3.A_{14}.C_2$

Order 85699747381248000: $C_2^{16}.A_{15}.C_2$ x 32, $C_2^{17}.A_{15}$ x 7

Order 68559797904998400: $(C_2\times C_2^3:A_4).C_2^{12}.A_{14}.C_2$ x 4, $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{14}$ x 3, $(C_2^2\times A_4).C_2^{12}.C_2^2.A_{14}.C_2$

Order 64274810535936000: $(C_2^{13}\times C_6).A_{15}.C_2$ x 4, $(C_2^{14}\times C_6).A_{15}$ x 3

Order 48971284217856000: $C_2^{17}.A_{13}.S_5$ x 6, $C_2^{18}.A_{13}.A_5$

Order 45706531936665600: $C_2^{16}.C_2^3.A_{14}.C_2$ x 6, $C_2^{17}.C_2^2.A_{14}.C_2$ x 4, $(C_2^2\times D_4).C_2^{12}.C_2^2.A_{14}.C_2$ x 2, $C_2^3.C_2^4.C_2^{12}.C_2.A_{14}$, $(C_2\times Q_8:C_2^2).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^2.D_4).C_2^{12}.C_2.A_{14}.C_2$

Order 45204262354944000: $C_2^{18}.A_{12}.A_6.C_2$

Order 42849873690624000: $C_2^{15}.A_{15}.C_2$ x 16, $C_2^{16}.A_{15}$ x 12, $C_2^{15}.A_{15}.C_2$ x 4

Order 34519618525593600: $C_2^{18}.A_9^2.C_4$

Order 34279898952499200: $(C_2\times A_4).C_2^{12}.C_2^2.A_{14}.C_2$ x 8, $(C_2^2\times A_4).C_2^{12}.C_2.A_{14}.C_2$ x 5, $(C_2^3\times A_4).C_2^{12}.A_{14}.C_2$ x 2, $(C_2\times \SL(2,3)).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^3:A_4).C_2^{12}.A_{14}$, $(C_2\times A_4).C_2^{12}.C_2^3.A_{14}$

Order 32137405267968000: $(C_2^{13}\times C_6).A_{15}$

Order 26369153040384000: $C_2^{18}.A_{11}.A_7.C_2$

Order 24485642108928000: $C_2^{16}.A_{13}.S_5$ x 4, $C_2^{17}.A_{13}.A_5$ x 3

Order 22853265968332800: $C_2^{16}.C_2^2.A_{14}.C_2$ x 19, $(C_2^2\times D_4).C_2^{12}.C_2.A_{14}.C_2$ x 9, $C_2^{17}.C_2.A_{14}.C_2$ x 4, $C_2^{16}.C_2^3.A_{14}$ x 3, $C_2^{15}.C_2^3.A_{14}.C_2$ x 3, $(C_2^{14}\times C_4).C_2^2.A_{14}.C_2$ x 3, $C_2^{17}.C_2^2.A_{14}$ x 2, $(C_2\times Q_8:C_2^2).C_2^{12}.A_{14}.C_2$ x 2, $(C_2\times C_2^2.D_4).C_2^{12}.A_{14}.C_2$ x 2, $C_2^{18}.A_{14}.C_2$, $C_2^{14}.C_2^4.A_{14}.C_2$, $(C_2^{13}\times C_4).C_2^3.A_{14}.C_2$, $(C_2\times D_4:C_2).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times D_4).C_2^{12}.C_2^2.A_{14}.C_2$, $(C_2\times C_2^2\wr C_2).C_2^{12}.C_2.A_{14}$, $(C_2\times C_2^2:C_4).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^2.D_4).C_2^{12}.C_2.A_{14}$

Order 22602131177472000: $C_2^{17}.A_{12}.A_6.C_2$ x 6, $C_2^{18}.A_{12}.A_6$

Order 21424936845312000: $C_2^{15}.A_{15}$ x 7, $C_2^{14}.A_{15}.C_2$ x 4

Order 19588513687142400: $(C_2\times C_2^3:A_4).C_2^{12}.C_2^2.A_{13}.C_2$

Order 19177565847552000: $C_2^{18}.A_{10}.A_8.C_2$

Order 17259809262796800: $C_2^{17}.A_9^2.C_4$ x 2, $C_2^{18}.A_9.A_9.C_2$

Order 17139949476249600: $(C_2\times A_4).C_2^{12}.C_2.A_{14}.C_2$ x 26, $(C_2^2\times A_4).C_2^{12}.A_{14}.C_2$ x 8, $(C_2^2\times A_4).C_2^{12}.C_2.A_{14}$ x 6, $(C_2\times \SL(2,3)).C_2^{12}.A_{14}.C_2$ x 2, $(C_2\times A_4).C_2^{12}.C_2^2.A_{14}$ x 2, $(C_2^{15}\times C_6).A_{14}.C_2$, $(C_2\times \SL(2,3)).C_2^{12}.C_2.A_{14}$

Order 16323761405952000: $(C_2\times C_2^4:D_5).C_2^{12}.C_2.A_{13}.C_2$

Order 13184576520192000: $C_2^{17}.A_{11}.A_7.C_2$ x 6, $C_2^{18}.A_{11}.A_7$

Order 12242821054464000: $C_2^{16}.A_{13}.A_5$

Order 11426632984166400: $C_2^{16}.C_2.A_{14}.C_2$ x 40, $C_2^{17}.A_{14}.C_2$ x 27, $C_2^{15}.C_2^2.A_{14}.C_2$ x 21, $(C_2^{14}\times C_4).C_2.A_{14}.C_2$ x 13, $C_2^{16}.C_2^2.A_{14}$ x 9, $(C_2^{15}\times C_4).A_{14}.C_2$ x 9, $(C_2^2\times D_4).C_2^{12}.A_{14}.C_2$ x 5, $C_2^{14}.C_2^3.A_{14}.C_2$ x 4, $(C_2^2\times D_4).C_2^{12}.C_2.A_{14}$ x 3, $(C_2^{14}\times C_4).C_2^2.A_{14}$ x 2, $(C_2\times C_2^2:C_4).C_2^{12}.C_2.A_{14}$ x 2, $(C_2\times C_2^2:C_4).C_2^{12}.A_{14}.C_2$ x 2, $C_2^{18}.A_{14}$, $C_2^{17}.C_2.A_{14}$, $C_2^{15}.C_2^3.A_{14}$, $(C_2^{13}\times C_4).C_2^2.A_{14}.C_2$, $(C_2\times D_4:C_2).C_2^{12}.A_{14}.C_2$, $(C_2\times D_4).C_2^{12}.C_2.A_{14}.C_2$, $(C_2\times C_2^2\wr C_2).C_2^{12}.A_{14}$

Order 11301065588736000: $C_2^{16}.A_{12}.A_6.C_2$ x 4, $C_2^{17}.A_{12}.A_6$ x 3

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

Order 9794256843571200: $C_2^3:A_4.C_2^{12}.C_2^2.A_{13}.C_2$ x 8, $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{13}.C_2$ x 3, $C_2\wr A_4.C_2^{12}.C_2.A_{13}.C_2$ x 2, $C_2^3:A_4.C_2^{12}.C_2^3.A_{13}$, $C_2\wr A_4.C_2.C_2^{12}.A_{13}.C_2$, $(C_2\times A_4).C_2^{12}.C_2^4.A_{13}.C_2$

Order 9588782923776000: $C_2^{17}.A_{10}.A_8.C_2$ x 6, $C_2^{18}.A_{10}.A_8$

Order 8629904631398400: $C_2^{17}.A_9.A_9.C_2$ x 4, $C_2^{16}.A_9^2.C_4$ x 2, $C_2^{18}.A_9.A_9$

Order 8569974738124800: $(C_2^{14}\times C_6).A_{14}.C_2$ x 14, $(C_2\times A_4).C_2^{12}.A_{14}.C_2$ x 12, $(C_2\times A_4).C_2^{12}.C_2.A_{14}$ x 8, $(C_2^2\times A_4).C_2^{12}.A_{14}$ x 2, $(C_2^{15}\times C_6).A_{14}$, $(C_2\times \SL(2,3)).C_2^{12}.A_{14}$

Order 8161880702976000: $C_2^4:C_5.C_2^{12}.C_2^2.A_{13}.C_2$ x 4, $C_2^4:D_5.C_2^{12}.C_2.A_{13}.C_2$ x 2, $(C_2\times C_2^4:D_5).C_2^{12}.C_2.A_{13}$

Order 7534043725824000: $C_2^{18}.A_{12}.S_5$ x 2

Order 6592288260096000: $C_2^{16}.A_{11}.A_7.C_2$ x 4, $C_2^{17}.A_{11}.A_7$ x 3

Order 6529504562380800: $C_2^{16}.C_2^4.A_{13}.C_2$

Order 6402373705728000: $C_2.A_{18}$

Order 6121410527232000: $C_2^{14}.A_{13}.S_5$

Order 5713316492083200: $C_2^{16}.A_{14}.C_2$ x 126, $(C_2^{14}\times C_4).A_{14}.C_2$ x 44, $C_2^{15}.C_2.A_{14}.C_2$ x 21, $C_2^{14}.C_2^2.A_{14}.C_2$ x 12, $C_2^{17}.A_{14}$ x 11, $C_2^{16}.C_2.A_{14}$ x 11, $C_2^{15}.C_2^2.A_{14}$ x 11, $(C_2^{13}\times C_4).C_2^2.A_{14}$ x 6, $(C_2^{14}\times C_4).C_2.A_{14}$ x 4, $(C_2^{15}\times C_4).A_{14}$ x 3, $(C_2^{13}\times C_4).C_2.A_{14}.C_2$ x 3, $(C_2\times D_4).C_2^{12}.A_{14}.C_2$ x 3, $C_2^{14}.C_2^3.A_{14}$, $(C_2\times Q_8).C_2^{12}.A_{14}.C_2$, $(C_2\times D_4:C_2).C_2^{12}.A_{14}$, $(C_2\times C_2^2:C_4).C_2^{12}.A_{14}$

Order 5650532794368000: $C_2^{16}.A_{12}.A_6$

Order 4897128421785600: $C_2^3:A_4.C_2^{12}.C_2.A_{13}.C_2$ x 17, $(C_2\times A_4).C_2^{12}.C_2^3.A_{13}.C_2$ x 11, $C_2^3:A_4.C_2^{12}.C_2^2.A_{13}$ x 6, $C_2\wr A_4.C_2^{12}.A_{13}.C_2$ x 6, $(C_2^2\times A_4).C_2^{12}.C_2^2.A_{13}.C_2$ x 5, $(C_2\times C_2^3:A_4).C_2^{12}.A_{13}.C_2$ x 5, $\GL(2,\mathbb{Z}/4).C_2^{12}.C_2.A_{13}.C_2$ x 4, $A_4:C_4.C_2^{12}.C_2^2.A_{13}.C_2$ x 3, $A_4.C_2^{12}.C_2^4.A_{13}.C_2$ x 3, $(C_2\times S_4).C_2^{12}.C_2^2.A_{13}.C_2$ x 3, $(C_2^2\times S_4).C_2^{12}.C_2.A_{13}.C_2$, $(C_2\times C_2^3:A_4).C_2^{12}.C_2.A_{13}$, $(C_2\times A_4).C_2^{12}.C_2^4.A_{13}$

Order 4794391461888000: $C_2^{16}.A_{10}.A_8.C_2$ x 4, $C_2^{17}.A_{10}.A_8$ x 3

Order 4520426235494400: $A_4^2.C_2^3.C_2^{10}.C_2^3.A_{12}.C_2$

Order 4314952315699200: $C_2^{16}.A_9.A_9.C_2$ x 3, $C_2^{17}.A_9.A_9$ x 2

Order 4284987369062400: $(C_2^{13}\times C_6).A_{14}.C_2$ x 28, $(C_2^{14}\times C_6).A_{14}$ x 7, $(C_2\times A_4).C_2^{12}.A_{14}$ x 2

Order 4080940351488000: $C_2^4:C_5.C_2^{12}.C_2.A_{13}.C_2$ x 3, $(C_2\times C_2^4:C_5).C_2^{12}.C_2.A_{13}$ x 3, $C_2^4:D_5.C_2^{12}.C_2.A_{13}$ x 2, $(C_2\times C_2^4:D_5).C_2^{12}.A_{13}$ x 2, $C_2^4:D_5.C_2^{12}.A_{13}.C_2$

Order 293534171136000: $C_2^{18}.A_6^3.S_4$

Order 24352464568320: $C_2^9.(C_2^{17}.A_9).C_2$

Order 4403012567040: $C_2^{12}.(C_3^6.(C_2^{10}.(C_2\times S_6)))$

Order 8589934592: $C_2^9.C_2^6.C_2^6.C_2^6.C_2^6$

Order 641728512: $C_2^{18}.\PSL(2,17)$

Order 262144: $C_2^{18}$

Order 131072: $C_2^{17}$

Order 6561: $C_3^4.C_3^4$

Order 125: $C_5^3$

Order 49: $C_7^2$

Order 17: $C_{17}$

Order 13: $C_{13}$

Order 11: $C_{11}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 839171926357180416000: $C_2^{18}.A_{18}$ ($C_1$)

Order 419585963178590208000: $C_2^{17}.A_{18}$ ($C_2$)

Order 262144: $C_2^{18}$ ($A_{18}$)

Order 131072: $C_2^{17}$ ($C_2.A_{18}$)

Order 2: $C_2$ ($C_2^{17}.A_{18}$)

Order 1: $C_1$ ($C_2^{18}.A_{18}$)

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

The $6208 \times 6208$ character table is not available for this group.

Rational character table

The $6148 \times 6148$ rational character table is not available for this group.