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## G = C2×C42⋊2S3order 192 = 26·3

### Direct product of C2 and C42⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C42⋊2S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×C42⋊2S3
 Lower central C3 — C6 — C2×C42⋊2S3
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C422S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 696 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C2×C42⋊C2, C422S3, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4, C2×C4×C12, S3×C22×C4, C2×C422S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C42⋊C2, C23×C4, C2×C4○D4, S3×C2×C4, C4○D12, S3×C23, C2×C42⋊C2, C422S3, S3×C22×C4, C2×C4○D12, C2×C422S3

Smallest permutation representation of C2×C422S3
On 96 points
Generators in S96
(1 57)(2 58)(3 59)(4 60)(5 94)(6 95)(7 96)(8 93)(9 13)(10 14)(11 15)(12 16)(17 76)(18 73)(19 74)(20 75)(21 77)(22 78)(23 79)(24 80)(25 40)(26 37)(27 38)(28 39)(29 44)(30 41)(31 42)(32 43)(33 48)(34 45)(35 46)(36 47)(49 64)(50 61)(51 62)(52 63)(53 68)(54 65)(55 66)(56 67)(69 92)(70 89)(71 90)(72 91)(81 88)(82 85)(83 86)(84 87)
(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 5 89 66)(2 6 90 67)(3 7 91 68)(4 8 92 65)(9 30 48 78)(10 31 45 79)(11 32 46 80)(12 29 47 77)(13 41 33 22)(14 42 34 23)(15 43 35 24)(16 44 36 21)(17 39 52 85)(18 40 49 86)(19 37 50 87)(20 38 51 88)(25 64 83 73)(26 61 84 74)(27 62 81 75)(28 63 82 76)(53 59 96 72)(54 60 93 69)(55 57 94 70)(56 58 95 71)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 39 44)(6 40 41)(7 37 42)(8 38 43)(9 58 73)(10 59 74)(11 60 75)(12 57 76)(21 66 85)(22 67 86)(23 68 87)(24 65 88)(25 30 95)(26 31 96)(27 32 93)(28 29 94)(33 90 49)(34 91 50)(35 92 51)(36 89 52)(45 72 61)(46 69 62)(47 70 63)(48 71 64)(53 84 79)(54 81 80)(55 82 77)(56 83 78)
(1 59)(2 60)(3 57)(4 58)(5 94)(6 95)(7 96)(8 93)(9 20)(10 17)(11 18)(12 19)(13 75)(14 76)(15 73)(16 74)(21 82)(22 83)(23 84)(24 81)(25 41)(26 42)(27 43)(28 44)(29 39)(30 40)(31 37)(32 38)(33 62)(34 63)(35 64)(36 61)(45 52)(46 49)(47 50)(48 51)(53 68)(54 65)(55 66)(56 67)(69 90)(70 91)(71 92)(72 89)(77 85)(78 86)(79 87)(80 88)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 1 ··· 1 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 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 C4○D4 C4×S3 C4○D12 kernel C2×C42⋊2S3 C42⋊2S3 C2×C4×Dic3 C2×Dic3⋊C4 C2×D6⋊C4 C2×C4×C12 S3×C22×C4 S3×C2×C4 C2×C42 C42 C22×C4 C2×C6 C2×C4 C22 # reps 1 8 1 2 2 1 1 16 1 4 3 8 8 16

Matrix representation of C2×C422S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 2 0 0 0 0 0 8
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 8 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,8,0,0,0,0,0,12] >;

C2×C422S3 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_2S_3
% in TeX

G:=Group("C2xC4^2:2S3");
// GroupNames label

G:=SmallGroup(192,1031);
// by ID

G=gap.SmallGroup(192,1031);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,1123,80,6278]);
// Polycyclic

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

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