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## G = C33⋊7(C2×C8)  order 432 = 24·33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊7(C2×C8)
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8 — C33⋊7(C2×C8)
 Lower central C33 — C33⋊7(C2×C8)
 Upper central C1 — C4

Generators and relations for C337(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, cd=dc, ece-1=c-1, de=ed >

Subgroups: 424 in 88 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊C8, C3×C3⋊S3, C32×C6, C322C8, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3⋊S33C8, C334C8, C12×C3⋊S3, C337(C2×C8)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C2×Dic3, C32⋊C4, C2×C3⋊C8, C2×C32⋊C4, C33⋊C4, C3⋊S33C8, C2×C33⋊C4, C337(C2×C8)

Smallest permutation representation of C337(C2×C8)
On 48 points
Generators in S48
(1 13 47)(2 14 48)(3 41 15)(4 42 16)(5 9 43)(6 10 44)(7 45 11)(8 46 12)(17 27 40)(18 33 28)(19 34 29)(20 30 35)(21 31 36)(22 37 32)(23 38 25)(24 26 39)
(2 48 14)(4 16 42)(6 44 10)(8 12 46)(17 40 27)(19 29 34)(21 36 31)(23 25 38)
(1 47 13)(2 14 48)(3 41 15)(4 16 42)(5 43 9)(6 10 44)(7 45 11)(8 12 46)(17 40 27)(18 28 33)(19 34 29)(20 30 35)(21 36 31)(22 32 37)(23 38 25)(24 26 39)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B ··· 3G 4A 4B 4C 4D 6A 6B ··· 6G 6H 6I 8A ··· 8H 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 3 3 ··· 3 4 4 4 4 6 6 ··· 6 6 6 8 ··· 8 12 12 12 ··· 12 12 12 size 1 1 9 9 2 4 ··· 4 1 1 9 9 2 4 ··· 4 18 18 27 ··· 27 2 2 4 ··· 4 18 18

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 C8 S3 D6 Dic3 Dic3 C3⋊C8 C32⋊C4 C2×C32⋊C4 C33⋊C4 C3⋊S3⋊3C8 C2×C33⋊C4 C33⋊7(C2×C8) kernel C33⋊7(C2×C8) C33⋊4C8 C12×C3⋊S3 C32×C12 C6×C3⋊S3 C3×C3⋊S3 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 8 1 1 1 1 4 2 2 4 4 4 8

Matrix representation of C337(C2×C8) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 54 0 0 0 0 8 0 19 0 0 0 0 8 0 0 0 0 0 0 64
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 67 48 0 0 0 1 6 48 0 0 0 0 64 0 0 0 0 0 0 8
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 8 0 19 19 0 0 0 8 54 19 0 0 0 0 64 0 0 0 0 0 0 64
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 45 0 0 0 1 0 0 28 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 10 0 0 0 0 10 0 0 0 0 0 0 0 67 67 51 51 0 0 67 6 51 51 0 0 0 63 67 6 0 0 63 0 6 6

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,54,0,8,0,0,0,0,19,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,67,6,64,0,0,0,48,48,0,8],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,19,54,64,0,0,0,19,19,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,45,0,0,1,0,0,0,28,1,0],[0,10,0,0,0,0,10,0,0,0,0,0,0,0,67,67,0,63,0,0,67,6,63,0,0,0,51,51,67,6,0,0,51,51,6,6] >;

C337(C2×C8) in GAP, Magma, Sage, TeX

C_3^3\rtimes_7(C_2\times C_8)
% in TeX

G:=Group("C3^3:7(C2xC8)");
// GroupNames label

G:=SmallGroup(432,635);
// by ID

G=gap.SmallGroup(432,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,64,58,2804,298,2693,1027,14118]);
// 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,c*d=d*c,e*c*e^-1=c^-1,d*e=e*d>;
// generators/relations

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