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## G = (C22×C8)⋊7S3order 192 = 26·3

### 3rd semidirect product of C22×C8 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C22×C8)⋊7S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×C4○D12 — (C22×C8)⋊7S3
 Lower central C3 — C2×C6 — (C22×C8)⋊7S3
 Upper central C1 — C2×C4 — C22×C8

Generators and relations for (C22×C8)⋊7S3
G = < a,b,c,d,e | a2=b2=c8=d3=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc4, cd=dc, ede=d-1 >

Subgroups: 408 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C2×C24, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, (C22×C8)⋊C2, D6⋊C8, C2×C4.Dic3, C22×C24, C2×C4○D12, (C22×C8)⋊7S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8○D4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, (C22×C8)⋊C2, C8○D12, C2×D6⋊C4, (C22×C8)⋊7S3

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C4×S3 C8○D4 C8○D12 kernel (C22×C8)⋊7S3 D6⋊C8 C2×C4.Dic3 C22×C24 C2×C4○D12 C2×Dic6 C2×D12 C2×C3⋊D4 C22×C8 C2×C12 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 2 2 4 1 4 2 1 2 4 4 2 8 16

Matrix representation of (C22×C8)⋊7S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 43 13 0 0 60 30
,
 66 59 0 0 14 7 0 0 0 0 22 0 0 0 0 22
,
 72 72 0 0 1 0 0 0 0 0 72 72 0 0 1 0
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 72 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,43,60,0,0,13,30],[66,14,0,0,59,7,0,0,0,0,22,0,0,0,0,22],[72,1,0,0,72,0,0,0,0,0,72,1,0,0,72,0],[1,72,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

(C22×C8)⋊7S3 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes_7S_3
% in TeX

G:=Group("(C2^2xC8):7S3");
// GroupNames label

G:=SmallGroup(192,669);
// by ID

G=gap.SmallGroup(192,669);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,422,58,136,6278]);
// Polycyclic

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

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