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## G = C4×C32⋊2C8order 288 = 25·32

### Direct product of C4 and C32⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C32⋊2C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C4×C32⋊2C8
 Lower central C32 — C4×C32⋊2C8
 Upper central C1 — C2×C4

Generators and relations for C4×C322C8
G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 272 in 74 conjugacy classes, 30 normal (16 characteristic)
C1, C2 [×3], C3 [×2], C4 [×2], C4 [×4], C22, C6 [×6], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], C42, C2×C8 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×2], C4×C8, C3⋊Dic3 [×4], C3×C12 [×2], C62, C4×Dic3 [×2], C322C8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×C3⋊Dic3, C2×C322C8 [×2], C4×C322C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], C42, C2×C8 [×2], C4×C8, C32⋊C4, C322C8 [×2], C2×C32⋊C4, C3⋊S33C8, C4×C32⋊C4, C2×C322C8, C4×C322C8

Smallest permutation representation of C4×C322C8
On 96 points
Generators in S96
(1 47 22 15)(2 48 23 16)(3 41 24 9)(4 42 17 10)(5 43 18 11)(6 44 19 12)(7 45 20 13)(8 46 21 14)(25 75 55 87)(26 76 56 88)(27 77 49 81)(28 78 50 82)(29 79 51 83)(30 80 52 84)(31 73 53 85)(32 74 54 86)(33 69 57 89)(34 70 58 90)(35 71 59 91)(36 72 60 92)(37 65 61 93)(38 66 62 94)(39 67 63 95)(40 68 64 96)
(2 74 68)(4 70 76)(6 78 72)(8 66 80)(10 34 26)(12 28 36)(14 38 30)(16 32 40)(17 90 88)(19 82 92)(21 94 84)(23 86 96)(42 58 56)(44 50 60)(46 62 52)(48 54 64)
(1 73 67)(2 74 68)(3 69 75)(4 70 76)(5 77 71)(6 78 72)(7 65 79)(8 66 80)(9 33 25)(10 34 26)(11 27 35)(12 28 36)(13 37 29)(14 38 30)(15 31 39)(16 32 40)(17 90 88)(18 81 91)(19 82 92)(20 93 83)(21 94 84)(22 85 95)(23 86 96)(24 89 87)(41 57 55)(42 58 56)(43 49 59)(44 50 60)(45 61 51)(46 62 52)(47 53 63)(48 54 64)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,47,22,15)(2,48,23,16)(3,41,24,9)(4,42,17,10)(5,43,18,11)(6,44,19,12)(7,45,20,13)(8,46,21,14)(25,75,55,87)(26,76,56,88)(27,77,49,81)(28,78,50,82)(29,79,51,83)(30,80,52,84)(31,73,53,85)(32,74,54,86)(33,69,57,89)(34,70,58,90)(35,71,59,91)(36,72,60,92)(37,65,61,93)(38,66,62,94)(39,67,63,95)(40,68,64,96), (2,74,68)(4,70,76)(6,78,72)(8,66,80)(10,34,26)(12,28,36)(14,38,30)(16,32,40)(17,90,88)(19,82,92)(21,94,84)(23,86,96)(42,58,56)(44,50,60)(46,62,52)(48,54,64), (1,73,67)(2,74,68)(3,69,75)(4,70,76)(5,77,71)(6,78,72)(7,65,79)(8,66,80)(9,33,25)(10,34,26)(11,27,35)(12,28,36)(13,37,29)(14,38,30)(15,31,39)(16,32,40)(17,90,88)(18,81,91)(19,82,92)(20,93,83)(21,94,84)(22,85,95)(23,86,96)(24,89,87)(41,57,55)(42,58,56)(43,49,59)(44,50,60)(45,61,51)(46,62,52)(47,53,63)(48,54,64), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (1,47,22,15)(2,48,23,16)(3,41,24,9)(4,42,17,10)(5,43,18,11)(6,44,19,12)(7,45,20,13)(8,46,21,14)(25,75,55,87)(26,76,56,88)(27,77,49,81)(28,78,50,82)(29,79,51,83)(30,80,52,84)(31,73,53,85)(32,74,54,86)(33,69,57,89)(34,70,58,90)(35,71,59,91)(36,72,60,92)(37,65,61,93)(38,66,62,94)(39,67,63,95)(40,68,64,96), (2,74,68)(4,70,76)(6,78,72)(8,66,80)(10,34,26)(12,28,36)(14,38,30)(16,32,40)(17,90,88)(19,82,92)(21,94,84)(23,86,96)(42,58,56)(44,50,60)(46,62,52)(48,54,64), (1,73,67)(2,74,68)(3,69,75)(4,70,76)(5,77,71)(6,78,72)(7,65,79)(8,66,80)(9,33,25)(10,34,26)(11,27,35)(12,28,36)(13,37,29)(14,38,30)(15,31,39)(16,32,40)(17,90,88)(18,81,91)(19,82,92)(20,93,83)(21,94,84)(22,85,95)(23,86,96)(24,89,87)(41,57,55)(42,58,56)(43,49,59)(44,50,60)(45,61,51)(46,62,52)(47,53,63)(48,54,64), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([(1,47,22,15),(2,48,23,16),(3,41,24,9),(4,42,17,10),(5,43,18,11),(6,44,19,12),(7,45,20,13),(8,46,21,14),(25,75,55,87),(26,76,56,88),(27,77,49,81),(28,78,50,82),(29,79,51,83),(30,80,52,84),(31,73,53,85),(32,74,54,86),(33,69,57,89),(34,70,58,90),(35,71,59,91),(36,72,60,92),(37,65,61,93),(38,66,62,94),(39,67,63,95),(40,68,64,96)], [(2,74,68),(4,70,76),(6,78,72),(8,66,80),(10,34,26),(12,28,36),(14,38,30),(16,32,40),(17,90,88),(19,82,92),(21,94,84),(23,86,96),(42,58,56),(44,50,60),(46,62,52),(48,54,64)], [(1,73,67),(2,74,68),(3,69,75),(4,70,76),(5,77,71),(6,78,72),(7,65,79),(8,66,80),(9,33,25),(10,34,26),(11,27,35),(12,28,36),(13,37,29),(14,38,30),(15,31,39),(16,32,40),(17,90,88),(18,81,91),(19,82,92),(20,93,83),(21,94,84),(22,85,95),(23,86,96),(24,89,87),(41,57,55),(42,58,56),(43,49,59),(44,50,60),(45,61,51),(46,62,52),(47,53,63),(48,54,64)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 8A ··· 8P 12A ··· 12H order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 4 4 1 1 1 1 9 ··· 9 4 ··· 4 9 ··· 9 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C4 C8 C8 C32⋊C4 C32⋊2C8 C2×C32⋊C4 C3⋊S3⋊3C8 C4×C32⋊C4 kernel C4×C32⋊2C8 C4×C3⋊Dic3 C2×C32⋊2C8 C32⋊2C8 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C3×C12 C2×C4 C4 C22 C2 C2 # reps 1 1 2 8 2 2 8 8 2 4 2 4 4

Matrix representation of C4×C322C8 in GL5(𝔽73)

 46 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 27
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 72 72
,
 1 0 0 0 0 0 72 72 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 72 72
,
 63 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 3 31 0 0 0 28 70 0 0

G:=sub<GL(5,GF(73))| [46,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,72],[63,0,0,0,0,0,0,0,3,28,0,0,0,31,70,0,1,0,0,0,0,0,1,0,0] >;

C4×C322C8 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C4xC3^2:2C8");
// GroupNames label

G:=SmallGroup(288,423);
// by ID

G=gap.SmallGroup(288,423);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,100,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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