Abstract group 139968000.a: $A_6\wr C_3$

• Label: 139968000.a
• Order: \( 2^{9} \cdot 3^{7} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{7} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$A_6\wr C_3$
Order:	\(139968000\)\(\medspace = 2^{9} \cdot 3^{7} \cdot 5^{3} \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(1119744000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \)
Composition factors:	$C_3$, $A_6$ x 3
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	20	30	60
Elements	1	97335	790640	2418120	3048624	13035600	20736000	3713040	28987200	45135360	12674880	3110400	6220800	139968000
Conjugacy classes	1	3	12	7	10	12	4	10	16	28	14	8	8	133
Divisions	1	3	11	7	5	11	2	5	15	13	7	4	4	88
Autjugacy classes	1	3	6	6	5	5	1	4	6	6	5	1	1	50

Dimension	1	2	15	24	27	30	48	75	120	125	135	150	192	216	240	243	250	270	300	375	384	432	480	512	600	675	729	750	960	1000	1024	1080	1200	1215	1350	1458	1500	1536	1728	1920	1944	2000	2048	2160	2400	2430	2700	3072	3456	3840	3888	4320	4800
Irr. complex chars.	3	0	2	2	1	1	0	4	8	6	4	4	4	4	4	1	0	2	1	2	0	0	0	6	8	4	3	4	8	3	0	8	8	2	4	0	2	2	4	4	2	0	0	4	2	1	1	0	0	0	0	0	0	133
Irr. rational chars.	1	1	2	0	1	1	1	4	0	2	4	4	0	0	4	1	2	2	1	2	2	2	2	0	0	4	1	4	0	1	1	0	4	2	4	1	2	0	0	4	0	1	1	4	4	1	1	1	2	2	1	2	1	88

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$18$
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 $18$ $\langle(1,7,15,5,12,14,4,9,18,6,10,17,3,8,13)(2,11,16), (1,7,17,2,12,16,5,10,18)(3,9,13,4,11,14,6,8,15)\rangle$
Transitive group:	18T967	30T4344	45T3916		more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(A_6.A_6.A_6)$ . $C_3$				more information

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 3 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	not computed
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 19200 (order at least 7290) are shown, as well as all normal subgroups of any index.

Order 139968000: $A_6\wr C_3$

Order 46656000: $A_6.A_6.A_6$

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

Order 4665600: $C_3:S_3.C_2\times A_6^2$

Order 3110400: $A_6^2\times S_4$ x 2

Order 2332800: $C_3:S_3.A_6.A_6$

Order 1555200: $A_4.A_6.A_6$ x 2

Order 1296000: $A_5^2\times A_6$ x 3, $D_5\times A_6^2$, $A_6\times A_5^2$

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

Order 1036800: $D_4.A_6.A_6$

Order 777600: $C_3:S_3.C_2.A_5.A_6$ x 3, $S_3\times A_6^2$ x 2, $C_3:S_3.C_2.A_6.A_5$

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

Order 518400: $S_4\times A_5\times A_6$ x 8, $C_2^2.A_6.A_6$ x 2, $C_4.A_6.A_6$

Order 466560: $C_3^4.C_4^2.A_6$

Order 388800: $C_3:S_3.A_5.A_6$ x 3, $C_3\times A_6^2$ x 2, $C_3:S_3.A_6.A_5$

Order 311040: $(C_3:S_3\times S_4).C_2.A_6$ x 4

Order 259200: $A_4\times A_5\times A_6$ x 8, $C_2\times A_6^2$

Order 233280: $C_3^4:C_2^2.C_2.A_6$ x 3

Order 216000: $A_5^3$ x 4, $D_5.A_5.A_6$ x 3, $D_5.A_6.A_5$

Order 207360: $A_6.S_4^2$ x 4

Order 194400: $C_3^2.A_5.A_6$ x 3, $C_3^2.A_6.A_5$

Order 172800: $D_4\times A_5\times A_6$ x 4

Order 155520: $(C_3^2\times A_4):C_4.A_6$ x 4, $(C_3:S_3\times S_4).A_6$ x 4, $(A_4\times C_3:S_3.C_2).A_6$ x 4

Order 139968: $C_3^6:C_4\wr C_3$

Order 129600: $S_3\times A_5\times A_6$ x 8, $A_6^2$ x 5, $C_3:S_3.C_2.A_5.A_5$ x 4, $(D_5\times C_3:S_3.C_2).A_6$ x 2

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

Order 108000: $(C_5\times A_5).A_6$ x 3, $C_5.A_6.A_5$

Order 103680: $A_4\times S_4\times A_6$ x 8, $A_6\times \PSOPlus(4,3)$ x 4, $(D_4\times C_3:S_3.C_2).A_6$ x 2

Order 86400: $S_4\times A_5^2$ x 8, $C_2^2\times A_5\times A_6$ x 8, $C_4\times A_5\times A_6$ x 4, $D_5\times S_4\times A_6$ x 4

Order 77760: $C_3^2:S_4.A_6$ x 4, $(C_3^2\times S_4).A_6$ x 4, $(A_4\times C_3:S_3).A_6$ x 4, $C_3^2:C_4\times S_3\times A_6$ x 4, $C_3^4.C_4^2.A_5$ x 2

Order 69984: $C_3^6:(C_4^2:C_6)$

Order 69120: $D_4\times S_4\times A_6$ x 4

Order 64800: $A_6\times \GL(2,4)$ x 8, $C_3:S_3.A_5.A_5$ x 4, $C_3^2:(C_5:C_4).A_6$ x 2, $(D_5\times C_3:S_3).A_6$ x 2, $(C_5\times C_3:S_3.C_2).A_6$ x 2

Order 58320: $(C_3^2\times C_3:S_3).A_6$ x 2, $C_3:(C_3^3:C_2).A_6$

Order 51840: $C_3^2:C_4\times S_4\times A_5$ x 8, $S_3\times S_4\times A_6$ x 8, $C_3:S_3.D_4.A_6$ x 4, $(C_2^2\times C_3:S_3.C_2).A_6$ x 4, $A_4^2\times A_6$ x 4, $(D_4\times C_3:S_3).A_6$ x 2, $(C_4\times C_3:S_3.C_2).A_6$ x 2, $(C_3\times C_{12}):C_4.A_6$ x 2

Order 46656: $C_3^6.C_4^3$

Order 43200: $A_4\times A_5^2$ x 8, $D_5\times A_4\times A_6$ x 4, $C_5:S_4\times A_6$ x 4, $C_5\times S_4\times A_6$ x 4, $C_2\times A_5\times A_6$ x 4

Order 41472: $S_4\wr C_3$ x 2

Order 38880: $C_3^4:C_2^2.C_2.A_5$ x 5, $C_3^3:C_4\times A_6$ x 4, $C_3:S_3^2\times A_6$ x 4, $C_3^2:C_{12}\times A_6$ x 4, $(C_3^2\times A_4).A_6$ x 3, $C_3^4.(C_2\times C_4).A_5$, $C_3\times (C_3\times A_4).A_6$

Order 36000: $D_5\times A_5^2$ x 4, $D_5^2.A_6$

Order 34992: $C_3^6.(C_4\times A_4)$, $C_3^5:D_6.A_4$

Order 34560: $A_6\times \GL(2,\mathbb{Z}/4)$ x 8, $A_5\times S_4^2$ x 8, $C_2^2\times S_4\times A_6$ x 8, $D_4\times A_4\times A_6$ x 4, $C_2^2:S_4\times A_6$ x 4, $C_4\times S_4\times A_6$ x 4, $C_4:S_4\times A_6$ x 4

Order 32400: $C_3^2.A_5.A_5$ x 4, $C_3:D_{15}.A_6$ x 2, $(C_5\times C_3:S_3).A_6$ x 2, $(C_3^2\times D_5).A_6$ x 2

Order 31104: $C_6^2.C_3^3.C_2^3.C_2^2$, $A_4.C_3^4.C_2^3.C_2^2$

Order 29160: $C_3^4.A_6$

Order 28800: $D_4\times A_5^2$ x 4, $D_4\times D_5\times A_6$ x 2

Order 25920: $C_3:S_4\times A_6$ x 16, $C_6^2:\GL(2,4):C_4$ x 8, $C_6^2:C_{12}\times A_5$ x 8, $S_4\times \GL(2,4):S_3$ x 8, $C_3\times S_4\times A_6$ x 8, $S_3\times A_4\times A_6$ x 8, $C_2\times C_3^2:C_4\times A_6$ x 8, $C_6^2:C_2.A_6$ x 4, $(C_2^2\times C_3:S_3).A_6$ x 4, $C_3:D_{12}.A_6$ x 2, $(C_4\times C_3:S_3).A_6$ x 2, $(C_3^2\times D_4).A_6$ x 2

Order 23328: $C_3^6.C_2^3.C_2^2$ x 3

Order 23040: $D_4^2\times A_6$

Order 21600: $A_5\times A_6$ x 20, $S_3\times A_5^2$ x 8, $(D_5\times C_3:S_3.C_2).A_5$ x 4, $S_3\times D_5\times A_6$ x 4, $C_5\times A_4\times A_6$ x 4

Order 20736: $C_3^2:C_4\times S_4^2$ x 4, $A_4^3:A_4$ x 2

Order 19440: $C_3^4.(C_4\times A_5)$ x 12, $C_3^2:S_3\times A_6$ x 8, $S_3\times C_3^2\times A_6$ x 4, $C_3^2:C_6\times A_6$ x 4, $C_3^4:C_2^2.A_5$ x 2

Order 18000: $C_5\times A_5^2$ x 4, $C_5\times D_5.A_6$, $C_5:D_5.A_6$, $(C_5\times D_5).A_6$

Order 17496: $C_3^6.(C_2\times A_4)$

Order 17280: $A_4\times S_4\times A_5$ x 16, $C_2\times S_4\times A_6$ x 12, $A_5\times \PSOPlus(4,3)$ x 8, $C_2^2\times A_4\times A_6$ x 8, $\GL(2,4):D_{12}:C_4$ x 4, $C_4\times A_4\times A_6$ x 4, $C_2^2:A_4\times A_6$ x 4, $A_4:C_4\times A_6$ x 4, $S_3\times D_4\times A_6$ x 4

Order 16200: $(C_3\times C_{15}).A_6$ x 2

Order 15552: $A_4.C_3^4.C_4^2$ x 5, $C_6^2.C_3^3.C_2^3.C_2$ x 2, $C_6^2.C_3^3.C_2.C_2^3$ x 2, $C_3^4.A_4.C_4^2$ x 2, $A_4.C_3^4.C_2^3.C_2$ x 2, $C_6^2.C_3^3.C_4^2$

Order 14400: $D_5\times S_4\times A_5$ x 8, $C_2^2\times A_5^2$ x 8, $C_2\times D_{10}\times A_6$ x 4, $C_4\times A_5^2$ x 4, $C_5:D_4\times A_6$ x 4, $C_5\times D_4\times A_6$ x 2, $D_{20}\times A_6$ x 2, $C_4\times D_5\times A_6$ x 2

Order 13824: $S_4^3$ x 4

Order 12960: $C_3\times A_4\times A_6$ x 16, $C_3^2:C_4\times A_6$ x 14, $C_6^2:C_6\times A_5$ x 8, $A_5\times C_3^2:S_4$ x 8, $C_3\times S_4\times \GL(2,4)$ x 8, $\GL(2,4):S_3.D_6$ x 8, $C_6:S_3\times A_6$ x 6, $C_6^2.A_6$ x 4, $S_3^2\times A_6$ x 4, $C_3^2:C_4.A_6$ x 2, $(C_3\times C_{12}).A_6$ x 2, $C_5.C_3^4:C_2^2.C_2^3$

Order 11664: $C_3^6.C_4^2$ x 8, $C_3^6.C_2^3.C_2$ x 2, $C_3^6:C_4^2$ x 2, $C_3^6:(C_2^2\times C_4)$

Order 11520: $D_4\times S_4\times A_5$ x 8, $C_4:D_4\times A_6$ x 4, $C_2^2\times D_4\times A_6$ x 4, $C_2^2\wr C_2\times A_6$ x 4, $C_4\times D_4\times A_6$ x 2, $C_4:D_4\times A_6$

Order 10800: $A_5\times \GL(2,4)$ x 8, $C_3^2:(C_5:C_4).A_5$ x 4, $(D_5\times C_3:S_3).A_5$ x 4, $(C_5\times C_3:S_3.C_2).A_5$ x 4, $D_{15}\times A_6$ x 4, $C_3\times D_5\times A_6$ x 4, $C_5\times S_3\times A_6$ x 4

Order 10368: $C_3:S_3.S_4^2$ x 8, $C_6^2:C_{12}\times S_4$ x 8, $C_3:S_3.S_4^2$ x 4, $C_6^2:C_{12}:S_4$ x 4, $C_3:S_3\times S_4^2$ x 4, $A_4^3:C_6$ x 2, $C_3^4.C_2^4.C_2^3$

Order 9720: $C_3^3\times A_6$ x 8, $C_3^4.(C_2\times A_5)$ x 6

Order 9000: $C_5^2.A_6$

Order 8748: $C_3^6.A_4$, $C_3^6.C_{12}$

Order 8640: $S_4\times A_6$ x 28, $S_3\times S_4\times A_5$ x 16, $C_6^2:C_4\times A_5$ x 8, $A_5\times A_4^2$ x 8, $C_3:D_4\times A_6$ x 8, $C_2\times D_6\times A_6$ x 8, $A_5\times C_6^2:C_4$ x 8, $C_2\times A_4\times A_6$ x 4, $(C_{12}\times \GL(2,4)):C_4$ x 4, $C_3^2:C_4^2\times A_5$ x 4, $C_6^2:C_{12}:D_{10}$ x 4, $C_4\times S_3\times A_6$ x 4, $D_{12}\times A_6$ x 4, $C_{12}:D_6\times A_5$ x 4, $C_3\times D_4\times A_6$ x 4

Order 7776: $A_4.C_3^4.C_2^2.C_2$ x 20, $C_6^2.C_3^3.C_2^2.C_2$ x 7, $C_3^4.A_4.C_2^2.C_2$ x 6, $C_2^2.C_3^5.C_2^2.C_2$ x 3, $S_3\times C_3^4:C_4^2$ x 2, $C_6^2.C_3^3.C_2^3$, $C_2^2.C_3^5.C_2^3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 139968000: $A_6\wr C_3$ ($C_1$)

Order 46656000: $A_6.A_6.A_6$ ($C_3$)

Order 1: $C_1$ ($A_6\wr C_3$)

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 14 larger groups in the database.

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

Character theory

Complex character table

See the $133 \times 133$ 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 $88 \times 88$ rational character table.