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## G = C33⋊M4(2)  order 432 = 24·33

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

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, C322C8, C322C8, S3×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, C62.C4, C3×C322C8, C334C8, 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

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