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## G = C33⋊12M4(2)  order 432 = 24·33

### 2nd semidirect product of C33 and M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊12M4(2)
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8 — C33⋊12M4(2)
 Lower central C33 — C32×C6 — C33⋊12M4(2)
 Upper central C1 — C2 — C22

Generators and relations for C3312M4(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=c-1, ce=ec, ede=d5 >

Subgroups: 392 in 92 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C4.Dic3, C32×C6, C32×C6, C322C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C62.C4, C334C8, C6×C3⋊Dic3, C3312M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C2×Dic3, C32⋊C4, C4.Dic3, C2×C32⋊C4, C33⋊C4, C62.C4, C2×C33⋊C4, C3312M4(2)

Permutation representations of C3312M4(2)
On 24 points - transitive group 24T1287
Generators in S24
(1 22 10)(2 11 23)(3 12 24)(4 17 13)(5 18 14)(6 15 19)(7 16 20)(8 21 9)
(1 10 22)(3 24 12)(5 14 18)(7 20 16)
(1 22 10)(2 11 23)(3 24 12)(4 13 17)(5 18 14)(6 15 19)(7 20 16)(8 9 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)

G:=sub<Sym(24)| (1,22,10)(2,11,23)(3,12,24)(4,17,13)(5,18,14)(6,15,19)(7,16,20)(8,21,9), (1,10,22)(3,24,12)(5,14,18)(7,20,16), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)>;

G:=Group( (1,22,10)(2,11,23)(3,12,24)(4,17,13)(5,18,14)(6,15,19)(7,16,20)(8,21,9), (1,10,22)(3,24,12)(5,14,18)(7,20,16), (1,22,10)(2,11,23)(3,24,12)(4,13,17)(5,18,14)(6,15,19)(7,20,16)(8,9,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23) );

G=PermutationGroup([[(1,22,10),(2,11,23),(3,12,24),(4,17,13),(5,18,14),(6,15,19),(7,16,20),(8,21,9)], [(1,10,22),(3,24,12),(5,14,18),(7,20,16)], [(1,22,10),(2,11,23),(3,24,12),(4,13,17),(5,18,14),(6,15,19),(7,20,16),(8,9,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23)]])

G:=TransitiveGroup(24,1287);

42 conjugacy classes

 class 1 2A 2B 3A 3B ··· 3G 4A 4B 4C 6A 6B 6C 6D ··· 6U 8A 8B 8C 8D 12A 12B 12C 12D order 1 2 2 3 3 ··· 3 4 4 4 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 size 1 1 2 2 4 ··· 4 9 9 18 2 2 2 4 ··· 4 54 54 54 54 18 18 18 18

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + - + + - image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 M4(2) C4.Dic3 C32⋊C4 C2×C32⋊C4 C33⋊C4 C62.C4 C2×C33⋊C4 C33⋊12M4(2) kernel C33⋊12M4(2) C33⋊4C8 C6×C3⋊Dic3 C3×C3⋊Dic3 C3×C62 C2×C3⋊Dic3 C3⋊Dic3 C3⋊Dic3 C62 C33 C32 C2×C6 C6 C22 C3 C2 C1 # reps 1 2 1 2 2 1 1 1 1 2 4 2 2 4 4 4 8

Matrix representation of C3312M4(2) in GL4(𝔽7) generated by

 3 2 4 3 4 5 5 6 3 3 6 1 0 0 0 1
,
 0 5 2 6 0 2 0 2 3 3 6 1 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 3 6 1 0 3 3 1 4 4 3 5 5 2 2 5 3
,
 0 6 3 2 6 0 4 2 0 0 6 0 0 0 0 1
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[3,3,4,2,6,3,3,2,1,1,5,5,0,4,5,3],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1] >;

C3312M4(2) in GAP, Magma, Sage, TeX

C_3^3\rtimes_{12}M_4(2)
% in TeX

G:=Group("C3^3:12M4(2)");
// GroupNames label

G:=SmallGroup(432,640);
// by ID

G=gap.SmallGroup(432,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,58,2804,298,2693,1027,14118]);
// 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=c^-1,c*e=e*c,e*d*e=d^5>;
// generators/relations

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