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## G = C2×C4⋊D12order 192 = 26·3

### Direct product of C2 and C4⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C4⋊D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C4⋊D12
 Lower central C3 — C2×C6 — C2×C4⋊D12
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C4⋊D12
G = < a,b,c,d | a2=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1656 in 498 conjugacy classes, 159 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, D4, C23, C23, C12, D6, C2×C6, C2×C6, C42, C22×C4, C2×D4, C24, D12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C41D4, C22×D4, C4×C12, C2×D12, C2×D12, C22×C12, S3×C23, C2×C41D4, C4⋊D12, C2×C4×C12, C22×D12, C2×C4⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C41D4, C22×D4, C2×D12, S3×C23, C2×C41D4, C4⋊D12, C22×D12, C2×C4⋊D12

Smallest permutation representation of C2×C4⋊D12
On 96 points
Generators in S96
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 49)(37 81)(38 82)(39 83)(40 84)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 85)(71 86)(72 87)
(1 33 75 92)(2 34 76 93)(3 35 77 94)(4 36 78 95)(5 25 79 96)(6 26 80 85)(7 27 81 86)(8 28 82 87)(9 29 83 88)(10 30 84 89)(11 31 73 90)(12 32 74 91)(13 58 43 65)(14 59 44 66)(15 60 45 67)(16 49 46 68)(17 50 47 69)(18 51 48 70)(19 52 37 71)(20 53 38 72)(21 54 39 61)(22 55 40 62)(23 56 41 63)(24 57 42 64)
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 72)(12 71)(13 85)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 90)(21 89)(22 88)(23 87)(24 86)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)(49 77)(50 76)(51 75)(52 74)(53 73)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4L 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 ··· 12 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 kernel C2×C4⋊D12 C4⋊D12 C2×C4×C12 C22×D12 C2×C42 C2×C12 C42 C22×C4 C2×C4 # reps 1 8 1 6 1 12 4 3 24

Matrix representation of C2×C4⋊D12 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 10 10 0 0 0 3 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 3 7 0 0 0 10 10

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

C2×C4⋊D12 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_{12}
% in TeX

G:=Group("C2xC4:D12");
// GroupNames label

G:=SmallGroup(192,1034);
// by ID

G=gap.SmallGroup(192,1034);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,80,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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