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## G = C33⋊5(C2×C8)  order 432 = 24·33

### 2nd semidirect product of C33 and C2×C8 acting via C2×C8/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊5(C2×C8)
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8(C2×C4) — C33⋊5(C2×C8)
 Lower central C33 — C33⋊5(C2×C8)
 Upper central C1 — C2

Generators and relations for C335(C2×C8)
G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, ac=ca, dad=a-1, eae-1=ab-1, bc=cb, dbd=b-1, ebe-1=a-1b-1, dcd=c-1, ce=ec, de=ed >

Subgroups: 864 in 96 conjugacy classes, 20 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, Dic3, C12, D6, C2×C8, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8, C33⋊C2, C32×C6, C322C8, C322C8, C6.D6, C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C3⋊S33C8, C3×C322C8, C334C8, C338(C2×C4), C335(C2×C8)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, C32⋊C4, S3×C8, C2×C32⋊C4, C3⋊S33C8, S3×C32⋊C4, C335(C2×C8)

Permutation representations of C335(C2×C8)
On 24 points - transitive group 24T1308
Generators in S24
(1 9 21)(2 10 22)(3 23 11)(4 24 12)(5 13 17)(6 14 18)(7 19 15)(8 20 16)
(2 22 10)(4 12 24)(6 18 14)(8 16 20)
(1 21 9)(2 22 10)(3 23 11)(4 24 12)(5 17 13)(6 18 14)(7 19 15)(8 20 16)
(1 5)(2 6)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1308);

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 27 27 2 4 4 8 8 3 3 9 9 2 4 4 8 8 9 9 9 9 27 27 27 27 12 12 12 12 18 18 18 18 18 18

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 C4×S3 S3×C8 C32⋊C4 C2×C32⋊C4 C3⋊S3⋊3C8 S3×C32⋊C4 C33⋊5(C2×C8) kernel C33⋊5(C2×C8) C3×C32⋊2C8 C33⋊4C8 C33⋊8(C2×C4) C32×Dic3 C2×C33⋊C2 C33⋊C2 C32⋊2C8 C3⋊Dic3 C3×C6 C32 Dic3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 8 1 1 2 4 2 2 4 2 2

Matrix representation of C335(C2×C8) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 72 71 72 72 0 0 1 1 0 0 0 0 0 1 1 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 72 0 72 0 0 0 72 72 0
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 10 10 10 0 0 0 0 0 63 10 0 0 0 0 0 63 0 0 0 63 0 63

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,1,71,1,1,0,0,0,72,0,1,0,0,1,72,0,0],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,72,72,0,0,0,0,0,72,0,0,0,0,72,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,10,0,0,0,0,0,10,0,0,63,0,0,10,63,0,0,0,0,0,10,63,63] >;

C335(C2×C8) in GAP, Magma, Sage, TeX

C_3^3\rtimes_5(C_2\times C_8)
% in TeX

G:=Group("C3^3:5(C2xC8)");
// GroupNames label

G:=SmallGroup(432,571);
// by ID

G=gap.SmallGroup(432,571);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,36,58,1411,298,1356,1027,14118]);
// Polycyclic

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

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