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## G = C22×C8⋊S3order 192 = 26·3

### Direct product of C22 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C22×C4 — C22×C8⋊S3
 Lower central C3 — C6 — C22×C8⋊S3
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C22×C8⋊S3
G = < a,b,c,d,e | a2=b2=c8=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >

Subgroups: 600 in 298 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C22×C8, C22×C8, C2×M4(2), C23×C4, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C22×M4(2), C2×C8⋊S3, C22×C3⋊C8, C22×C24, S3×C22×C4, C22×C8⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C24, C4×S3, C22×S3, C2×M4(2), C23×C4, C8⋊S3, S3×C2×C4, S3×C23, C22×M4(2), C2×C8⋊S3, S3×C22×C4, C22×C8⋊S3

Smallest permutation representation of C22×C8⋊S3
On 96 points
Generators in S96
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)(49 80)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)(41 95)(42 96)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(65 79)(66 80)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(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 86 75)(2 87 76)(3 88 77)(4 81 78)(5 82 79)(6 83 80)(7 84 73)(8 85 74)(9 47 50)(10 48 51)(11 41 52)(12 42 53)(13 43 54)(14 44 55)(15 45 56)(16 46 49)(17 28 70)(18 29 71)(19 30 72)(20 31 65)(21 32 66)(22 25 67)(23 26 68)(24 27 69)(33 64 94)(34 57 95)(35 58 96)(36 59 89)(37 60 90)(38 61 91)(39 62 92)(40 63 93)
(1 20)(2 17)(3 22)(4 19)(5 24)(6 21)(7 18)(8 23)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(25 77)(26 74)(27 79)(28 76)(29 73)(30 78)(31 75)(32 80)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)(49 92)(50 89)(51 94)(52 91)(53 96)(54 93)(55 90)(56 95)(65 86)(66 83)(67 88)(68 85)(69 82)(70 87)(71 84)(72 81)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 6A ··· 6G 8A ··· 8H 8I ··· 8P 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 6 6 6 6 2 1 ··· 1 6 6 6 6 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C22×C8⋊S3 C2×C8⋊S3 C22×C3⋊C8 C22×C24 S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C22×C8 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 1 1 12 2 2 1 6 1 8 6 2 16

Matrix representation of C22×C8⋊S3 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 45 36 0 0 0 54 28
,
 1 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 72 26 0 0 0 0 1

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,45,54,0,0,0,36,28],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,72,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,26,1] >;

C22×C8⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_8\rtimes S_3
% in TeX

G:=Group("C2^2xC8:S3");
// GroupNames label

G:=SmallGroup(192,1296);
// by ID

G=gap.SmallGroup(192,1296);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,80,102,6278]);
// Polycyclic

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

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