Abstract group 559872000.o: $C_2\times A_6\wr S_3$

• Label: 559872000.o
• Order: \( 2^{11} \cdot 3^{7} \cdot 5^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{7} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2\times A_6\wr S_3$
Order:	\(559872000\)\(\medspace = 2^{11} \cdot 3^{7} \cdot 5^{3} \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(2239488000\)\(\medspace = 2^{13} \cdot 3^{7} \cdot 5^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $A_6$ x 3
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	18	20	24	30	40	60
Elements	1	294031	790640	18249840	3048624	48807440	26438400	20736000	69883344	81302400	45135360	20736000	67340160	15552000	101122560	27993600	12441600	559872000
Conjugacy classes	1	11	10	20	9	46	6	2	43	28	18	2	26	4	42	4	8	280
Divisions	1	11	10	20	5	46	6	2	23	28	10	2	14	4	22	2	4	210
Autjugacy classes	1	9	6	16	5	21	3	1	18	14	6	1	12	1	10	1	2	127

Dimension	1	2	15	24	27	30	48	75	125	150	192	240	243	250	270	300	375	384	432	480	512	540	600	675	729	750	864	960	1000	1024	1200	1215	1350	1458	1500	1536	1728	1920	1944	2000	2048	2160	2400	2430	2700	3072	3456	3840	3888	4320	4800	8640
Irr. complex chars.	4	2	8	8	4	4	0	8	8	2	8	8	4	4	4	8	8	2	4	4	8	2	16	8	4	8	0	16	4	4	4	8	2	2	10	8	8	12	8	2	0	8	16	4	8	0	2	2	0	4	0	0	280
Irr. rational chars.	4	2	8	0	4	4	4	8	8	2	0	0	4	4	4	8	8	6	0	4	0	2	0	8	4	8	2	2	4	4	8	8	2	2	10	0	0	12	0	2	2	0	2	4	8	4	6	6	4	4	8	2	210

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	15	15	15
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $20$ $\langle(1,3)(2,5,9,4,6,10,14,15)(7,8,12,17)(11,16,18,13), (1,2,4,7,11,15,3,6,5,8,9,13,17,18,10)(12,14,16)(19,20)\rangle$
Transitive group:	36T88318				more information
Direct product:	$C_2$ $\, \times\, $ $(A_6\wr S_3)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_6^3$ . $D_6$	$(A_6\wr S_3)$ . $C_2$ (2)	$C_2$ . $(A_6\wr S_3)$	$(A_6\wr C_3)$ . $C_2^2$	all 6

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 9 normal subgroups (7 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $A_6\wr C_3$
Frattini:	a subgroup isomorphic to $C_1$
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 8748 (order at least 64000) are shown, as well as all normal subgroups of any index.

Order 559872000: $C_2\times A_6\wr S_3$

Order 279936000: $A_6\wr S_3$ x 2, $C_2\times A_6\wr C_3$

Order 186624000: $C_2.A_6.A_6^2.C_2$

Order 139968000: $A_6\wr C_3$

Order 93312000: $A_6.A_6^2.C_2$ x 2, $C_2.A_6.A_6.A_6$

Order 46656000: $A_6^3$

Order 31104000: $(C_2\times A_5).A_6^2.C_2$ x 2

Order 18662400: $(C_2\times C_3:S_3.C_2).A_6^2.C_2$

Order 15552000: $A_5.A_6^2.C_2$ x 4, $(C_2\times A_5).A_6.A_6$ x 2

Order 12441600: $(C_2\times S_4).A_6^2.C_2$ x 2

Order 9331200: $C_3:S_3.C_2.A_6^2.C_2$ x 4, $(C_2\times C_3:S_3).A_6^2.C_2$ x 2, $(C_2\times C_3:S_3.C_2).A_6.A_6$

Order 7776000: $A_6^2\times A_5$ x 2

Order 6220800: $S_4.A_6^2.C_2$ x 8, $(C_2\times A_4).A_6^2.C_2$ x 4, $(C_2\times S_4).A_6.A_6$ x 2

Order 5184000: $C_2.A_6.A_5^2.C_2$ x 2, $D_{10}.A_6^2.C_2$

Order 4665600: $C_3:S_3.A_6^2.C_2$ x 4, $C_3:S_3.C_2\times A_6^2$ x 2, $(C_3\times C_6).A_6^2.C_2$ x 2, $C_3^2.A_6^2:C_4$ x 2, $(C_2\times C_3:S_3).A_6.A_6$

Order 4147200: $(C_2\times D_4).A_6^2.C_2$

Order 3110400: $A_4.A_6^2.C_2$ x 8, $A_6^2\times S_4$ x 4, $D_6.A_6^2.C_2$ x 2, $(C_2\times A_4).A_6.A_6$ x 2

Order 2592000: $D_5.A_6^2.C_2$ x 4, $A_6.A_5^2.C_2$ x 4, $(C_2\times A_5).A_5.A_6$ x 3, $C_{10}.A_6^2.C_2$ x 2, $A_5^3.D_6$ x 2, $D_{10}.A_6.A_6$

Order 2332800: $C_3^2.A_6^2.C_2$ x 4, $C_3:S_3.A_6.A_6$ x 2, $(C_3\times C_6).A_6.A_6$

Order 2073600: $D_4.A_6^2.C_2$ x 8, $C_2^3.A_6^2.C_2$ x 4, $(C_2\times C_4).A_6^2.C_2$ x 2, $(C_2\times D_4).A_6.A_6$

Order 1866240: $C_3^4.(C_2\times C_4\wr C_2).A_6$

Order 1555200: $A_6^2.D_6$ x 8, $C_6.A_6^2.C_2$ x 4, $D_6.A_6.A_6$ x 2, $C_3:S_3.(C_2^2\times A_5).A_6$ x 2, $A_4.A_6.A_6$ x 2

Order 1296000: $C_5.A_6^2.C_2$ x 4, $A_5\wr S_3$ x 4, $A_5^2\times A_6$ x 3, $A_6^2\times D_5$ x 2, $A_5^3.C_6$ x 2, $C_{10}.A_6.A_6$

Order 1166400: $C_3^2.A_6.A_6$

Order 1036800: $C_2^2.A_6^2.C_2$ x 12, $C_4.A_6^2.C_2$ x 8, $A_6^2.D_4$ x 8, $D_4.A_6.A_6$ x 4, $(C_2\times S_4).A_5.A_6$ x 4, $C_2^3.A_6.A_6$ x 2, $(C_2\times C_4).A_6.A_6$

Order 933120: $C_3^4.C_4\wr C_2.A_6$ x 4, $C_3^4.C_2^3.C_2^2.A_6$ x 2, $C_3^4.(C_2\times \OD_{16}).A_6$

Order 864000: $(C_2\times A_5).A_5^2.C_2$ x 4

Order 829440: $A_4^2.C_2^3.C_2.A_6$ x 2

Order 777600: $S_3\times A_6^2$ x 4, $A_6^2:C_6$ x 4, $A_6^2:S_3$ x 4, $C_6.A_6.A_6$ x 2, $C_3:S_3.C_2.A_6.A_5$ x 2, $C_3:S_3.C_2.A_5.A_6$ x 2, $(C_2\times C_3:S_3).A_6.A_5$, $(C_2\times C_3:S_3).A_5.A_6$

Order 648000: $A_5\wr C_3$ x 2, $C_5.A_6.A_6$

Order 622080: $(C_3:S_3\times S_4).C_2^2.A_6$, $(A_4\times C_3:S_3).C_2^3.A_6$

Order 559872: $C_3^6.C_4^3:D_6$

Order 518400: $A_6^2:C_2^2$ x 12, $S_4\times A_5\times A_6$ x 8, $C_2^2.A_6.A_6$ x 6, $(C_2\times A_4).A_5.A_6$ x 3, $C_4.A_6.A_6$ x 2, $C_2.A_6.A_6.C_2$ x 2, $(C_2\times C_3:S_3.C_2).A_5^2.C_2$ x 2, $A_6^2:C_4$ x 2, $(C_2\times A_4).A_6.A_5$, $A_6^2.C_2^2$

Order 466560: $C_3^4.C_2^3.C_2.A_6$ x 5, $C_3^4.\OD_{16}.A_6$ x 3, $C_3^4.D_4:C_2.A_6$ x 3, $C_3^4.D_4.C_2.A_6$ x 3, $C_3^4.C_4^2.A_6$ x 3, $C_3^4.(C_2\times Q_8).A_6$, $C_3^4.(C_2\times C_8).A_6$

Order 432000: $A_5^3:C_2$ x 8, $(C_2\times A_5).A_5.A_5$ x 4, $D_{10}.A_5.A_6$ x 2

Order 414720: $A_6\times A_4^2.D_4$ x 8, $A_4^2.C_2^3.A_6$ x 5, $A_4^2.C_2^2.C_2.A_6$ x 2

Order 388800: $C_3:S_3.A_5.A_6$ x 4, $(C_3\times C_6).A_5.A_6$ x 2, $C_3\times A_6^2$ x 2

Order 345600: $(C_2\times S_4).A_5^2.C_2$ x 4, $(C_2\times D_4).A_6.A_5$, $(C_2\times D_4).A_5.A_6$

Order 311040: $(C_3:S_3\times S_4).C_2.A_6$ x 6, $C_6^2.(C_4\times S_3).A_6$ x 2, $C_6^2.(C_2\times C_{12}).A_6$ x 2, $C_3^4.(C_2\times C_4\wr C_2).A_5$ x 2, $(A_4\times C_3:S_3).C_2^2.A_6$ x 2, $(C_3^2\times A_4).C_2^3.A_6$, $(C_3\times C_6\times S_4).C_2.A_6$

Order 279936: $C_3^6:C_4\wr S_3$ x 4, $C_3^6.(C_2^4.S_4)$, $C_3^6.(C_2^4.S_4)$, $C_3^6.C_4^3:C_6$

Order 259200: $C_2\times A_6^2$ x 9, $C_3:S_3.C_2.A_5^2.C_2$ x 8, $A_6.A_6.C_2$ x 4, $(C_2\times C_3:S_3).A_5^2.C_2$ x 4, $A_4\times A_5\times A_6$ x 4, $A_6\wr C_2$ x 4, $C_3:S_3.(C_2^2\times A_5).A_5$ x 3, $D_6.A_5.A_6$ x 2, $A_6\times S_6$ x 2, $D_6.A_6.A_5$, $C_2\times (S_3\times A_5).A_6$, $(C_5\times C_3:S_3).C_2^3.A_6$

Order 233280: $C_3^4:C_2^2.C_2.A_6$ x 9, $C_3^4.(C_2\times C_4).A_6$ x 8, $C_3^4.D_4.A_6$ x 7, $C_3^4.Q_8.A_6$ x 3, $C_3^4.C_8.A_6$ x 2, $C_3^4.C_2^3.A_6$ x 2

Order 216000: $A_5^3$ x 4, $A_5\times D_5\times A_6$ x 3, $C_{10}.A_6.A_5$, $C_{10}.A_5.A_6$, $A_5\times A_6\times D_5$

Order 207360: $A_6.S_4^2$ x 10, $A_4^2.C_2^2.A_6$ x 9, $A_6\times \POPlus(4,3)$ x 6, $A_4^2:C_4\times A_6$ x 4, $(C_2\times C_3:S_3).C_2^4.A_6$

Order 194400: $C_3^2.A_6.A_5$, $C_3^2.A_5.A_6$

Order 186624: $C_3^6.C_4^3.C_2^2$

Order 172800: $S_4\times \SOPlus(4,4)$ x 16, $(C_2\times A_4).A_5^2.C_2$ x 8, $D_4\times A_5\times A_6$ x 8, $(C_2\times S_4).A_5.A_5$ x 6, $C_2^3.A_6.A_5$ x 2, $(C_2^3\times A_5).A_6$ x 2, $(C_2\times D_5\times S_4).A_6$ x 2, $(C_2\times C_4\times A_5).A_6$, $(C_2\times C_4).A_6.A_5$

Order 165888: $S_4^3.D_6$ x 2

Order 155520: $C_3^4.C_4\wr C_2.A_5$ x 8, $(C_3:S_3\times S_4).A_6$ x 8, $C_3^4.C_2^3.C_2^2.A_5$ x 4, $(C_3^2\times A_4):C_4.A_6$ x 4, $(A_4\times C_3:S_3.C_2).A_6$ x 4, $C_3^4.(C_2\times \OD_{16}).A_5$ x 2, $(C_3\times C_6\times S_4).A_6$ x 2, $(C_2\times S_3\times C_3:S_3.C_2).A_6$ x 2, $(C_2\times C_3^2:S_4).A_6$ x 2, $(C_2\times A_4\times C_3:S_3).A_6$ x 2

Order 144000: $D_{10}.A_5^2.C_2$ x 2, $(C_2\times D_5\wr C_2).A_6$

Order 139968: $(C_3:S_3)^3.S_4$ x 4, $(C_3:S_3)^3.S_4$ x 2, $C_3^6:C_4\wr C_3$ x 2, $C_3^6:(C_4^2:D_6)$, $C_3^6:(C_4^2:D_6)$, $C_3^6.(C_2^4.A_4)$, $C_3^6.(C_2^4.D_6)$

Order 138240: $(A_5\times S_4^2).C_2^2$ x 4, $C_2^4:C_3.C_2^3.A_6$ x 2, $(C_2^3\times S_4).C_2.A_6$ x 2

Order 129600: $C_3:S_3.A_5^2.C_2$ x 8, $S_3\times A_5\times A_6$ x 8, $C_3:S_3.C_2.A_5.A_5$ x 6, $A_6^2$ x 5, $(D_5\times C_3:S_3.C_2).A_6$ x 4, $(C_6\times A_5).A_6$ x 4, $(C_3\times C_6).A_5^2.C_2$ x 4, $C_3^2:(A_5^2:C_4)$ x 4, $(C_2\times C_3:S_3).A_5.A_5$ x 3, $(C_{10}\times C_3:S_3.C_2).A_6$, $(C_2\times D_5\times C_3:S_3).A_6$, $(C_2\times C_3^2:(C_5:C_4)).A_6$

Order 124416: $C_3^4.A_4.C_4^2.C_2^3$ x 2

Order 116640: $C_3^4:C_2^2.A_6$ x 6, $C_3^2:(C_3:S_3.C_2).A_6$ x 6, $C_3^3:(C_3:C_4).A_6$ x 2, $(C_3^2\times C_3:S_3.C_2).A_6$ x 2, $(C_3\times C_6\times C_3:S_3).A_6$ x 2, $(C_2\times C_3:(C_3^3:C_2)).A_6$

Order 115200: $(C_2\times D_4).A_5^2.C_2$ x 2

Order 108000: $C_5.A_6.A_5$, $(C_5\times A_5).A_6$

Order 103680: $(D_4\times C_3:S_3.C_2).A_6$ x 8, $A_4\times S_4\times A_6$ x 8, $A_6\times \PSOPlus(4,3)$ x 6, $(A_4\times C_3:S_3).C_2^3.A_5$ x 4, $A_4\wr C_2\times A_6$ x 4, $D_6\times S_4\times A_6$ x 4, $(C_2\times A_4^2).A_6$ x 3, $(C_2^3\times C_3:S_3.C_2).A_6$ x 2, $(C_2\times C_3:S_3.D_4).A_6$ x 2, $(C_2\times D_4\times C_3:S_3).A_6$, $(C_2\times C_4\times C_3:S_3.C_2).A_6$, $(C_2\times (C_3\times C_{12}):C_4).A_6$

Order 93312: $C_3^6.C_4.C_4^2.C_2$ x 5, $C_3^6.C_4^3.C_2$ x 3, $C_3^6.C_4.C_2^2.C_2^3$ x 3, $C_3^6.C_4^2.C_2^3$, $C_3^6.C_4.C_2.C_2^4$, $C_3^6.C_2^3.C_2^4$, $C_3^6.(C_4\times \OD_{16}).C_2$

Order 92160: $D_4^2.C_2^2.A_6$

Order 86400: $C_2^2\times A_5\times A_6$ x 14, $S_4\times A_5^2$ x 12, $A_5^2:S_4$ x 8, $A_4\times \SOPlus(4,4)$ x 8, $D_5\times S_4\times A_6$ x 8, $(C_2\times A_4).A_5.A_5$ x 6, $C_4\times A_5\times A_6$ x 4, $D_6\times \SOPlus(4,4)$ x 4, $C_2.A_6.A_5.C_2$ x 2, $(C_{10}\times S_4).A_6$ x 2, $(C_2\times D_5\times A_4).A_6$ x 2, $(C_2\times C_5:S_4).A_6$ x 2, $(C_2\times A_5).A_6.C_2$ x 2, $C_2\times A_5\times S_6$ x 2, $C_2\times S_5\times A_6$ x 2

Order 82944: $A_4^3.(C_2\times S_4)$ x 8, $A_4^2.C_3^2.C_2^3.C_2^3$ x 2, $C_2\times A_4^3:S_4$ x 2, $C_2\times A_4^3:S_4$ x 2, $S_4^3.C_6$ x 2

Order 77760: $C_3^4.D_4.C_2.A_5$ x 11, $C_3^2:C_4\times S_3\times A_6$ x 8, $C_3^4.C_2^3.C_2.A_5$ x 7, $C_3^4.\OD_{16}.A_5$ x 6, $C_3^4.C_4^2.A_5$ x 6, $C_3^2:S_4.A_6$ x 4, $(C_3^2\times S_4).A_6$ x 4, $(A_4\times C_3:S_3).A_6$ x 4, $(C_2\times S_3\times C_3:S_3).A_6$ x 3, $C_3^4.(C_2^2\times C_4).A_5$ x 2, $C_3^4.(C_2\times Q_8).A_5$ x 2, $C_3^4.(C_2\times C_8).A_5$ x 2, $(C_6\times C_3:S_3.C_2).A_6$ x 2, $(C_3\times C_6\times A_4).A_6$ x 2, $(C_2\times C_3^2:(C_3:C_4)).A_6$ x 2, $C_3^4.D_4:C_2.A_5$, $C_3^4.(C_2\times D_4).A_5$, $C_2\times (S_3\times C_3:S_3).A_6$

Order 72000: $A_5^2:D_{10}$ x 8, $D_5\wr C_2.A_6$ x 4, $C_{10}.A_5^2.C_2$ x 3, $D_{10}.A_5.A_5$ x 2, $(C_2\times D_5^2).A_6$ x 2, $C_2\times D_5.A_5.A_5$, $C_2\times C_5.A_5^2.C_2$, $(C_2\times C_5:F_5).A_6$

Order 69984: $C_3^6:(C_4\times S_4)$ x 4, $C_3^5:D_6.S_4$ x 2, $C_3^5:D_6.S_4$ x 2, $C_3^6:(C_4^2:C_6)$ x 2, $C_3^6.A_4.C_2^3$, $C_3^6.C_4^2:C_6$, $(C_3:S_3)^3.D_6$, $(C_3:S_3)^3.C_{12}$

Order 69120: $A_5\times S_4\wr C_2$ x 16, $D_4\times S_4\times A_6$ x 16, $A_6\times C_2^2:S_4:C_2$ x 8, $(C_2\times A_5).S_4^2$ x 6, $(C_2\times C_2^2:S_4).A_6$ x 5, $(C_2^3\times S_4).A_6$ x 4, $(C_2\times \GL(2,\mathbb{Z}/4)).A_6$ x 4, $C_2\times A_4^2:C_4\times A_5$ x 4, $A_4^2:C_2^3\times A_5$ x 4, $(C_2\times D_4\times A_4).A_6$ x 2, $(C_2\times C_4\times S_4).A_6$ x 2, $(C_2\times C_4:S_4).A_6$ x 2, $(C_2\times C_2^2\wr C_2:C_3).A_6$ x 2

Order 64800: $C_3^2.A_5^2.C_2$ x 8, $C_3:S_3.A_5.A_5$ x 6, $(D_5\times C_3:S_3).A_6$ x 4, $A_6\times \GL(2,4)$ x 4, $(C_3\times C_6).A_5.A_5$ x 3, $C_3^2:(C_5:C_4).A_6$ x 2, $(C_5\times C_3:S_3.C_2).A_6$ x 2, $(C_{10}\times C_3:S_3).A_6$, $(C_3\times C_6\times D_5).A_6$, $(C_2\times C_3:D_{15}).A_6$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 559872000: $C_2\times A_6\wr S_3$ ($C_1$)

Order 279936000: $A_6\wr S_3$ ($C_2$) x 2, $C_2\times A_6\wr C_3$ ($C_2$)

Order 139968000: $A_6\wr C_3$ ($C_2^2$)

Order 93312000: $C_2.A_6.A_6.A_6$ ($S_3$)

Order 46656000: $A_6^3$ ($D_6$)

Order 2: $C_2$ ($A_6\wr S_3$)

Order 1: $C_1$ ($C_2\times A_6\wr S_3$)

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 5 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $280 \times 280$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $210 \times 210$ rational character table (warning: may be slow to load).