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## G = C4×C8⋊S3order 192 = 26·3

### Direct product of C4 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C42 — C4×C8⋊S3
 Lower central C3 — C6 — C4×C8⋊S3
 Upper central C1 — C42 — C4×C8

Generators and relations for C4×C8⋊S3
G = < a,b,c,d | a4=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 280 in 142 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, S3×C2×C4, C4×M4(2), C4×C3⋊C8, C24⋊C4, C4×C24, S3×C42, C2×C8⋊S3, C4×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, M4(2), C22×C4, C4×S3, C22×S3, C2×C42, C2×M4(2), C8⋊S3, S3×C2×C4, C4×M4(2), S3×C42, C2×C8⋊S3, C4×C8⋊S3

Smallest permutation representation of C4×C8⋊S3
On 96 points
Generators in S96
(1 65 75 15)(2 66 76 16)(3 67 77 9)(4 68 78 10)(5 69 79 11)(6 70 80 12)(7 71 73 13)(8 72 74 14)(17 28 52 84)(18 29 53 85)(19 30 54 86)(20 31 55 87)(21 32 56 88)(22 25 49 81)(23 26 50 82)(24 27 51 83)(33 61 93 45)(34 62 94 46)(35 63 95 47)(36 64 96 48)(37 57 89 41)(38 58 90 42)(39 59 91 43)(40 60 92 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)(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)
(1 43 18)(2 44 19)(3 45 20)(4 46 21)(5 47 22)(6 48 23)(7 41 24)(8 42 17)(9 93 87)(10 94 88)(11 95 81)(12 96 82)(13 89 83)(14 90 84)(15 91 85)(16 92 86)(25 69 35)(26 70 36)(27 71 37)(28 72 38)(29 65 39)(30 66 40)(31 67 33)(32 68 34)(49 79 63)(50 80 64)(51 73 57)(52 74 58)(53 75 59)(54 76 60)(55 77 61)(56 78 62)
(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 67)(10 72)(11 69)(12 66)(13 71)(14 68)(15 65)(16 70)(17 62)(18 59)(19 64)(20 61)(21 58)(22 63)(23 60)(24 57)(25 95)(26 92)(27 89)(28 94)(29 91)(30 96)(31 93)(32 90)(33 87)(34 84)(35 81)(36 86)(37 83)(38 88)(39 85)(40 82)(41 51)(42 56)(43 53)(44 50)(45 55)(46 52)(47 49)(48 54)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4L 4M ··· 4R 6A 6B 6C 8A ··· 8H 8I ··· 8P 12A ··· 12L 24A ··· 24P order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 6 6 2 1 ··· 1 6 ··· 6 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C4×C8⋊S3 C4×C3⋊C8 C24⋊C4 C4×C24 S3×C42 C2×C8⋊S3 C8⋊S3 C4×Dic3 S3×C2×C4 C4×C8 C42 C2×C8 C12 C8 C2×C4 C4 # reps 1 1 2 1 1 2 16 4 4 1 1 2 8 8 4 16

Matrix representation of C4×C8⋊S3 in GL5(𝔽73)

 27 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 27 0 0 0 0 0 46 71 0 0 0 13 27 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0
,
 1 0 0 0 0 0 1 0 0 0 0 46 72 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(73))| [27,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[27,0,0,0,0,0,46,13,0,0,0,71,27,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,1,46,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0] >;

C4×C8⋊S3 in GAP, Magma, Sage, TeX

C_4\times C_8\rtimes S_3
% in TeX

G:=Group("C4xC8:S3");
// GroupNames label

G:=SmallGroup(192,246);
// by ID

G=gap.SmallGroup(192,246);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,58,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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