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## G = C22×D4⋊S3order 192 = 26·3

### Direct product of C22 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C22×D4⋊S3
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C22×D12 — C22×D4⋊S3
 Lower central C3 — C6 — C12 — C22×D4⋊S3
 Upper central C1 — C23 — C22×C4 — C22×D4

Generators and relations for C22×D4⋊S3
G = < a,b,c,d,e,f | a2=b2=c4=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf=c-1, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 1000 in 338 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, D4, C23, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, D12, D12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22×C6, C22×C8, C2×D8, C22×D4, C22×D4, C2×C3⋊C8, D4⋊S3, C2×D12, C2×D12, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, C22×D8, C22×C3⋊C8, C2×D4⋊S3, C22×D12, D4×C2×C6, C22×D4⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C24, C3⋊D4, C22×S3, C2×D8, C22×D4, D4⋊S3, C2×C3⋊D4, S3×C23, C22×D8, C2×D4⋊S3, C22×C3⋊D4, C22×D4⋊S3

Smallest permutation representation of C22×D4⋊S3
On 96 points
Generators in S96
(1 60)(2 57)(3 58)(4 59)(5 51)(6 52)(7 49)(8 50)(9 55)(10 56)(11 53)(12 54)(13 63)(14 64)(15 61)(16 62)(17 67)(18 68)(19 65)(20 66)(21 71)(22 72)(23 69)(24 70)(25 75)(26 76)(27 73)(28 74)(29 79)(30 80)(31 77)(32 78)(33 83)(34 84)(35 81)(36 82)(37 87)(38 88)(39 85)(40 86)(41 91)(42 92)(43 89)(44 90)(45 95)(46 96)(47 93)(48 94)
(1 34)(2 35)(3 36)(4 33)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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 93)(2 96)(3 95)(4 94)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 82)(22 81)(23 84)(24 83)(25 91)(26 90)(27 89)(28 92)(29 87)(30 86)(31 85)(32 88)(33 70)(34 69)(35 72)(36 71)(37 79)(38 78)(39 77)(40 80)(41 75)(42 74)(43 73)(44 76)(45 58)(46 57)(47 60)(48 59)(49 67)(50 66)(51 65)(52 68)(53 63)(54 62)(55 61)(56 64)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 93 10)(6 94 11)(7 95 12)(8 96 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(57 61 66)(58 62 67)(59 63 68)(60 64 65)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 85 90)(82 86 91)(83 87 92)(84 88 89)
(1 34)(2 33)(3 36)(4 35)(5 77)(6 80)(7 79)(8 78)(9 73)(10 76)(11 75)(12 74)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)(25 53)(26 56)(27 55)(28 54)(29 49)(30 52)(31 51)(32 50)(57 83)(58 82)(59 81)(60 84)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 96)(70 95)(71 94)(72 93)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A ··· 6G 6H ··· 6O 8A ··· 8H 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 12 12 12 size 1 1 ··· 1 4 4 4 4 12 12 12 12 2 2 2 2 2 2 ··· 2 4 ··· 4 6 ··· 6 4 4 4 4

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 C3⋊D4 C3⋊D4 D4⋊S3 kernel C22×D4⋊S3 C22×C3⋊C8 C2×D4⋊S3 C22×D12 D4×C2×C6 C22×D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×C6 C2×C4 C23 C22 # reps 1 1 12 1 1 1 3 1 1 6 8 6 2 4

Matrix representation of C22×D4⋊S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 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 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 71 0 0 0 0 1 72
,
 43 13 0 0 0 0 60 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 16 0
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

C22×D4⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes S_3
% in TeX

G:=Group("C2^2xD4:S3");
// GroupNames label

G:=SmallGroup(192,1351);
// by ID

G=gap.SmallGroup(192,1351);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,1684,235,102,6278]);
// Polycyclic

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

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