C3^3:M4(2) - GroupNames

G = C₃³⋊M₄(2) order 432 = 2⁴·3³

1^st semidirect product of C₃³ and M₄(2) acting via M₄(2)/C₂=C₂×C₄

Aliases: C₃³⋊₁M₄(2), D₆.(C₃²⋊C₄), C₃²⋊₂C₈⋊₃S₃, C₃³⋊₄C₈⋊₆C₂, C₃⋊Dic₃.₂₅D₆, C₃³⋊₅C₄.₂C₄, C₃²⋊₉(C₈⋊S₃), C₃⋊₁(C₆².C₄), (S₃×C₃×C₆).₂C₄, C₂.₆(S₃×C₃²⋊C₄), C₆.₆(C₂×C₃²⋊C₄), (C₃×C₆).₃₁(C₄×S₃), (C₃×C₃²⋊₂C₈)⋊₆C₂, (S₃×C₃⋊Dic₃).₄C₂, (C₃²×C₆).₆(C₂×C₄), (C₃×C₃⋊Dic₃).₂₈C₂², SmallGroup(432,572)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — C₃³⋊M₄(2)

Chief series

C₁ — C₃ — C₃³ — C₃²×C₆ — C₃×C₃⋊Dic₃ — S₃×C₃⋊Dic₃ — C₃³⋊M₄(2)

Lower central

C₃³ — C₃²×C₆ — C₃³⋊M₄(2)

Upper central

C₁ — C₂

Generators and relations for C₃³⋊M₄(2)
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, ac=ca, dad^-1=ab^-1, ae=ea, bc=cb, dbd^-1=a^-1b^-1, be=eb, dcd^-1=ece=c^-1, ede=d⁵ >

Subgroups: 544 in 84 conjugacy classes, 18 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, D₆, C₂×C₆, M₄(2), C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, C₂×Dic₃, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃⋊Dic₃, S₃×C₆, C₆², C₈⋊S₃, S₃×C₃², C₃²×C₆, C₃²⋊₂C₈, C₃²⋊₂C₈, S₃×Dic₃, C₂×C₃⋊Dic₃, C₃×C₃⋊Dic₃, C₃³⋊₅C₄, S₃×C₃×C₆, C₆².C₄, C₃×C₃²⋊₂C₈, C₃³⋊₄C₈, S₃×C₃⋊Dic₃, C₃³⋊M₄(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₆, M₄(2), C₄×S₃, C₃²⋊C₄, C₈⋊S₃, C₂×C₃²⋊C₄, C₆².C₄, S₃×C₃²⋊C₄, C₃³⋊M₄(2)

Character table of C₃³⋊M₄(2)

class	1	2A	2B	3A	3B	3C	3D	3E	4A	4B	4C	6A	6B	6C	6D	6E	6F	6G	6H	6I	8A	8B	8C	8D	12A	12B	24A	24B	24C	24D
size	1	1	6	2	4	4	8	8	9	9	54	2	4	4	8	8	12	12	12	12	18	18	54	54	18	18	18	18	18	18
ρ₁	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	trivial
ρ₂	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	linear of order 2
ρ₃	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	linear of order 2
ρ₄	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	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	-1	-1	-1	1	1	1	1	1	1	1	1	1	i	-i	-i	i	-1	-1	i	-i	-i	i	linear of order 4
ρ₆	1	1	-1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	i	-i	i	-i	-1	-1	i	-i	-i	i	linear of order 4
ρ₇	1	1	1	1	1	1	1	1	-1	-1	-1	1	1	1	1	1	1	1	1	1	-i	i	i	-i	-1	-1	-i	i	i	-i	linear of order 4
ρ₈	1	1	-1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-i	i	-i	i	-1	-1	-i	i	i	-i	linear of order 4
ρ₉	2	2	0	-1	2	2	-1	-1	2	2	0	-1	2	2	-1	-1	0	0	0	0	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₀	2	2	0	-1	2	2	-1	-1	2	2	0	-1	2	2	-1	-1	0	0	0	0	-2	-2	0	0	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₁	2	-2	0	2	2	2	2	2	-2i	2i	0	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	complex lifted from M₄(2)
ρ₁₂	2	2	0	-1	2	2	-1	-1	-2	-2	0	-1	2	2	-1	-1	0	0	0	0	-2i	2i	0	0	1	1	i	-i	-i	i	complex lifted from C₄×S₃
ρ₁₃	2	2	0	-1	2	2	-1	-1	-2	-2	0	-1	2	2	-1	-1	0	0	0	0	2i	-2i	0	0	1	1	-i	i	i	-i	complex lifted from C₄×S₃
ρ₁₄	2	-2	0	2	2	2	2	2	2i	-2i	0	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	complex lifted from M₄(2)
ρ₁₅	2	-2	0	-1	2	2	-1	-1	2i	-2i	0	1	-2	-2	1	1	0	0	0	0	0	0	0	0	i	-i	2ζ₈³ζ₃+ζ₈³	2ζ₈ζ₃+ζ₈	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈⁷ζ₃+ζ₈⁷	complex lifted from C₈⋊S₃
ρ₁₆	2	-2	0	-1	2	2	-1	-1	-2i	2i	0	1	-2	-2	1	1	0	0	0	0	0	0	0	0	-i	i	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈³ζ₃+ζ₈³	2ζ₈ζ₃+ζ₈	complex lifted from C₈⋊S₃
ρ₁₇	2	-2	0	-1	2	2	-1	-1	2i	-2i	0	1	-2	-2	1	1	0	0	0	0	0	0	0	0	i	-i	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈⁵ζ₃+ζ₈⁵	2ζ₈ζ₃+ζ₈	2ζ₈³ζ₃+ζ₈³	complex lifted from C₈⋊S₃
ρ₁₈	2	-2	0	-1	2	2	-1	-1	-2i	2i	0	1	-2	-2	1	1	0	0	0	0	0	0	0	0	-i	i	2ζ₈ζ₃+ζ₈	2ζ₈³ζ₃+ζ₈³	2ζ₈⁷ζ₃+ζ₈⁷	2ζ₈⁵ζ₃+ζ₈⁵	complex lifted from C₈⋊S₃
ρ₁₉	4	4	-4	4	1	-2	-2	1	0	0	0	4	-2	1	1	-2	-1	-1	2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₂₀	4	4	4	4	-2	1	1	-2	0	0	0	4	1	-2	-2	1	-2	-2	1	1	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₂₁	4	4	4	4	1	-2	-2	1	0	0	0	4	-2	1	1	-2	1	1	-2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₂₂	4	4	-4	4	-2	1	1	-2	0	0	0	4	1	-2	-2	1	2	2	-1	-1	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂×C₃²⋊C₄
ρ₂₃	4	-4	0	4	1	-2	-2	1	0	0	0	-4	2	-1	-1	2	-3	3	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₆².C₄, Schur index 2
ρ₂₄	4	-4	0	4	-2	1	1	-2	0	0	0	-4	-1	2	2	-1	0	0	3	-3	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₆².C₄, Schur index 2
ρ₂₅	4	-4	0	4	1	-2	-2	1	0	0	0	-4	2	-1	-1	2	3	-3	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₆².C₄, Schur index 2
ρ₂₆	4	-4	0	4	-2	1	1	-2	0	0	0	-4	-1	2	2	-1	0	0	-3	3	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₆².C₄, Schur index 2
ρ₂₇	8	8	0	-4	2	-4	2	-1	0	0	0	-4	-4	2	-1	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₃×C₃²⋊C₄
ρ₂₈	8	8	0	-4	-4	2	-1	2	0	0	0	-4	2	-4	2	-1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₃×C₃²⋊C₄
ρ₂₉	8	-8	0	-4	-4	2	-1	2	0	0	0	4	-2	4	-2	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₃₀	8	-8	0	-4	2	-4	2	-1	0	0	0	4	4	-2	1	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₃³⋊M₄(2)
►On 48 points

Generators in S₄₈

(1 38 45)(2 39 46)(3 47 40)(4 48 33)(5 34 41)(6 35 42)(7 43 36)(8 44 37)(9 28 23)(10 24 29)(11 17 30)(12 31 18)(13 32 19)(14 20 25)(15 21 26)(16 27 22)

(2 46 39)(4 33 48)(6 42 35)(8 37 44)(9 23 28)(11 30 17)(13 19 32)(15 26 21)

(1 45 38)(2 39 46)(3 47 40)(4 33 48)(5 41 34)(6 35 42)(7 43 36)(8 37 44)(9 23 28)(10 29 24)(11 17 30)(12 31 18)(13 19 32)(14 25 20)(15 21 26)(16 27 22)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

(1 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 35)(10 40)(11 37)(12 34)(13 39)(14 36)(15 33)(16 38)(25 43)(26 48)(27 45)(28 42)(29 47)(30 44)(31 41)(32 46)

G:=sub<Sym(48)| (1,38,45)(2,39,46)(3,47,40)(4,48,33)(5,34,41)(6,35,42)(7,43,36)(8,44,37)(9,28,23)(10,24,29)(11,17,30)(12,31,18)(13,32,19)(14,20,25)(15,21,26)(16,27,22), (2,46,39)(4,33,48)(6,42,35)(8,37,44)(9,23,28)(11,30,17)(13,19,32)(15,26,21), (1,45,38)(2,39,46)(3,47,40)(4,33,48)(5,41,34)(6,35,42)(7,43,36)(8,37,44)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,43)(26,48)(27,45)(28,42)(29,47)(30,44)(31,41)(32,46)>;

G:=Group( (1,38,45)(2,39,46)(3,47,40)(4,48,33)(5,34,41)(6,35,42)(7,43,36)(8,44,37)(9,28,23)(10,24,29)(11,17,30)(12,31,18)(13,32,19)(14,20,25)(15,21,26)(16,27,22), (2,46,39)(4,33,48)(6,42,35)(8,37,44)(9,23,28)(11,30,17)(13,19,32)(15,26,21), (1,45,38)(2,39,46)(3,47,40)(4,33,48)(5,41,34)(6,35,42)(7,43,36)(8,37,44)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,43)(26,48)(27,45)(28,42)(29,47)(30,44)(31,41)(32,46) );

G=PermutationGroup([[(1,38,45),(2,39,46),(3,47,40),(4,48,33),(5,34,41),(6,35,42),(7,43,36),(8,44,37),(9,28,23),(10,24,29),(11,17,30),(12,31,18),(13,32,19),(14,20,25),(15,21,26),(16,27,22)], [(2,46,39),(4,33,48),(6,42,35),(8,37,44),(9,23,28),(11,30,17),(13,19,32),(15,26,21)], [(1,45,38),(2,39,46),(3,47,40),(4,33,48),(5,41,34),(6,35,42),(7,43,36),(8,37,44),(9,23,28),(10,29,24),(11,17,30),(12,31,18),(13,19,32),(14,25,20),(15,21,26),(16,27,22)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,35),(10,40),(11,37),(12,34),(13,39),(14,36),(15,33),(16,38),(25,43),(26,48),(27,45),(28,42),(29,47),(30,44),(31,41),(32,46)]])

Matrix representation of C₃³⋊M₄(2) ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	0	72	0
0	0	0	72	0	1
0	0	1	0	0	0
0	0	0	72	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	72	1	1
0	0	1	0	1	0
0	0	72	1	0	0
0	0	1	0	0	0

0	72	0	0	0	0
1	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

25	7	0	0	0	0
32	48	0	0	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	1	72	0	0
0	0	0	0	1	1

29	25	0	0	0	0
54	44	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,72,0,0,72,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,72,1,0,0,72,0,1,0,0,0,1,1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,32,0,0,0,0,7,48,0,0,0,0,0,0,72,0,1,0,0,0,0,0,72,0,0,0,0,1,0,1,0,0,0,0,0,1],[29,54,0,0,0,0,25,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃³⋊M₄(2) in GAP, Magma, Sage, TeX

C_3^3\rtimes M_4(2)

% in TeX

G:=Group("C3^3:M4(2)");

// GroupNames label

G:=SmallGroup(432,572);

// by ID

G=gap.SmallGroup(432,572);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,36,58,1411,298,1356,1027,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,a*e=e*a,b*c=c*b,d*b*d^-1=a^-1*b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;

// generators/relations

Export

Character table of C₃³⋊M₄(2) in TeX

G = C33⋊M4(2) order 432 = 24·33

1st semidirect product of C33 and M4(2) acting via M4(2)/C2=C2×C4

G = C₃³⋊M₄(2) order 432 = 2⁴·3³

1^st semidirect product of C₃³ and M₄(2) acting via M₄(2)/C₂=C₂×C₄