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## G = C2×C4⋊C4⋊S3order 192 = 26·3

### Direct product of C2 and C4⋊C4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C4⋊C4⋊S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×D6⋊C4 — C2×C4⋊C4⋊S3
 Lower central C3 — C2×C6 — C2×C4⋊C4⋊S3
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C4⋊C4⋊S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 600 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, S3×C23, C2×C422C2, C4⋊C4⋊S3, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, C6×C4⋊C4, C2×C4⋊C4⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C422C2, C2×C4○D4, C4○D12, D42S3, Q83S3, S3×C23, C2×C422C2, C4⋊C4⋊S3, C2×C4○D12, C2×D42S3, C2×Q83S3, C2×C4⋊C4⋊S3

Smallest permutation representation of C2×C4⋊C4⋊S3
On 96 points
Generators in S96
(1 80)(2 77)(3 78)(4 79)(5 93)(6 94)(7 95)(8 96)(9 31)(10 32)(11 29)(12 30)(13 88)(14 85)(15 86)(16 87)(17 35)(18 36)(19 33)(20 34)(21 40)(22 37)(23 38)(24 39)(25 64)(26 61)(27 62)(28 63)(41 49)(42 50)(43 51)(44 52)(45 68)(46 65)(47 66)(48 67)(53 83)(54 84)(55 81)(56 82)(57 91)(58 92)(59 89)(60 90)(69 73)(70 74)(71 75)(72 76)
(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 83 58)(2 96 84 57)(3 95 81 60)(4 94 82 59)(5 53 92 80)(6 56 89 79)(7 55 90 78)(8 54 91 77)(9 23 65 36)(10 22 66 35)(11 21 67 34)(12 24 68 33)(13 26 43 74)(14 25 44 73)(15 28 41 76)(16 27 42 75)(17 32 37 47)(18 31 38 46)(19 30 39 45)(20 29 40 48)(49 72 86 63)(50 71 87 62)(51 70 88 61)(52 69 85 64)
(1 87 35)(2 88 36)(3 85 33)(4 86 34)(5 27 32)(6 28 29)(7 25 30)(8 26 31)(9 96 61)(10 93 62)(11 94 63)(12 95 64)(13 18 77)(14 19 78)(15 20 79)(16 17 80)(21 82 49)(22 83 50)(23 84 51)(24 81 52)(37 53 42)(38 54 43)(39 55 44)(40 56 41)(45 90 73)(46 91 74)(47 92 75)(48 89 76)(57 70 65)(58 71 66)(59 72 67)(60 69 68)
(2 84)(4 82)(5 7)(6 91)(8 89)(9 72)(10 64)(11 70)(12 62)(13 38)(14 19)(15 40)(16 17)(18 43)(20 41)(21 86)(22 50)(23 88)(24 52)(25 32)(26 48)(27 30)(28 46)(29 74)(31 76)(33 85)(34 49)(35 87)(36 51)(37 42)(39 44)(45 75)(47 73)(54 77)(56 79)(57 94)(58 60)(59 96)(61 67)(63 65)(66 69)(68 71)(90 92)(93 95)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 D4⋊2S3 Q8⋊3S3 kernel C2×C4⋊C4⋊S3 C4⋊C4⋊S3 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×D6⋊C4 C6×C4⋊C4 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C6 C22 C22 C22 # reps 1 8 1 1 1 3 1 1 4 3 12 8 2 2

Matrix representation of C2×C4⋊C4⋊S3 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 12 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 10 0 0 0 0 0 5 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 1 0 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,12,5,0,0,0,0,0,10,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C2×C4⋊C4⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes C_4\rtimes S_3
% in TeX

G:=Group("C2xC4:C4:S3");
// GroupNames label

G:=SmallGroup(192,1071);
// by ID

G=gap.SmallGroup(192,1071);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,675,297,136,6278]);
// 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,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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