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## G = C2×C33⋊4C8order 432 = 24·33

### Direct product of C2 and C33⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C33⋊4C8
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8 — C2×C33⋊4C8
 Lower central C33 — C2×C33⋊4C8
 Upper central C1 — C22

Generators and relations for C2×C334C8
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >

Subgroups: 392 in 96 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C62, C62, C2×C3⋊C8, C32×C6, C32×C6, C322C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C3×C62, C2×C322C8, C334C8, C6×C3⋊Dic3, C2×C334C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C2×Dic3, C32⋊C4, C2×C3⋊C8, C322C8, C2×C32⋊C4, C33⋊C4, C2×C322C8, C334C8, C2×C33⋊C4, C2×C334C8

Smallest permutation representation of C2×C334C8
On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)
(1 13 17)(2 18 14)(3 19 15)(4 16 20)(5 9 21)(6 22 10)(7 23 11)(8 12 24)(25 42 33)(26 34 43)(27 35 44)(28 45 36)(29 46 37)(30 38 47)(31 39 48)(32 41 40)
(1 17 13)(3 15 19)(5 21 9)(7 11 23)(25 33 42)(27 44 35)(29 37 46)(31 48 39)
(1 17 13)(2 14 18)(3 19 15)(4 16 20)(5 21 9)(6 10 22)(7 23 11)(8 12 24)(25 33 42)(26 43 34)(27 35 44)(28 45 36)(29 37 46)(30 47 38)(31 39 48)(32 41 40)
(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)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (1,13,17)(2,18,14)(3,19,15)(4,16,20)(5,9,21)(6,22,10)(7,23,11)(8,12,24)(25,42,33)(26,34,43)(27,35,44)(28,45,36)(29,46,37)(30,38,47)(31,39,48)(32,41,40), (1,17,13)(3,15,19)(5,21,9)(7,11,23)(25,33,42)(27,44,35)(29,37,46)(31,48,39), (1,17,13)(2,14,18)(3,19,15)(4,16,20)(5,21,9)(6,10,22)(7,23,11)(8,12,24)(25,33,42)(26,43,34)(27,35,44)(28,45,36)(29,37,46)(30,47,38)(31,39,48)(32,41,40), (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( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (1,13,17)(2,18,14)(3,19,15)(4,16,20)(5,9,21)(6,22,10)(7,23,11)(8,12,24)(25,42,33)(26,34,43)(27,35,44)(28,45,36)(29,46,37)(30,38,47)(31,39,48)(32,41,40), (1,17,13)(3,15,19)(5,21,9)(7,11,23)(25,33,42)(27,44,35)(29,37,46)(31,48,39), (1,17,13)(2,14,18)(3,19,15)(4,16,20)(5,21,9)(6,10,22)(7,23,11)(8,12,24)(25,33,42)(26,43,34)(27,35,44)(28,45,36)(29,37,46)(30,47,38)(31,39,48)(32,41,40), (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([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40)], [(1,13,17),(2,18,14),(3,19,15),(4,16,20),(5,9,21),(6,22,10),(7,23,11),(8,12,24),(25,42,33),(26,34,43),(27,35,44),(28,45,36),(29,46,37),(30,38,47),(31,39,48),(32,41,40)], [(1,17,13),(3,15,19),(5,21,9),(7,11,23),(25,33,42),(27,44,35),(29,37,46),(31,48,39)], [(1,17,13),(2,14,18),(3,19,15),(4,16,20),(5,21,9),(6,10,22),(7,23,11),(8,12,24),(25,33,42),(26,43,34),(27,35,44),(28,45,36),(29,37,46),(30,47,38),(31,39,48),(32,41,40)], [(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 6C 6D ··· 6U 8A ··· 8H 12A 12B 12C 12D order 1 2 2 2 3 3 ··· 3 4 4 4 4 6 6 6 6 ··· 6 8 ··· 8 12 12 12 12 size 1 1 1 1 2 4 ··· 4 9 9 9 9 2 2 2 4 ··· 4 27 ··· 27 18 18 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 Dic3 D6 Dic3 C3⋊C8 C32⋊C4 C32⋊2C8 C2×C32⋊C4 C33⋊C4 C33⋊4C8 C2×C33⋊C4 kernel C2×C33⋊4C8 C33⋊4C8 C6×C3⋊Dic3 C3×C3⋊Dic3 C3×C62 C32×C6 C2×C3⋊Dic3 C3⋊Dic3 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 2 8 1 1 1 1 4 2 4 2 4 8 4

Matrix representation of C2×C334C8 in GL6(𝔽73)

 72 0 0 0 0 0 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 19 0 0 0 8 19 0 0 0 0 0 64 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 48 48 0 0 0 64 6 67 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 8 0 54 54 0 0 0 8 19 54 0 0 0 0 64 0 0 0 0 0 0 64
,
 0 22 0 0 0 0 22 0 0 0 0 0 0 0 67 6 0 0 0 0 67 67 0 0 0 0 0 63 6 67 0 0 10 0 6 6

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,19,64,0,0,0,19,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,48,6,1,0,0,0,48,67,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,54,19,64,0,0,0,54,54,0,64],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,67,67,0,10,0,0,6,67,63,0,0,0,0,0,6,6,0,0,0,0,67,6] >;

C2×C334C8 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_4C_8
% in TeX

G:=Group("C2xC3^3:4C8");
// GroupNames label

G:=SmallGroup(432,639);
// by ID

G=gap.SmallGroup(432,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,2804,298,2693,1027,14118]);
// Polycyclic

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

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