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## G = C2×C4×D12order 192 = 26·3

### Direct product of C2×C4 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4×D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C4×D12
 Lower central C3 — C6 — C2×C4×D12
 Upper central C1 — C22×C4 — 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, bd=db, dcd=c-1 >

Subgroups: 1048 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, S3×C2×C4, C2×D12, C22×Dic3, C22×C12, S3×C23, C2×C4×D4, C4×D12, C2×C4⋊Dic3, C2×D6⋊C4, C2×C4×C12, S3×C22×C4, C22×D12, C2×C4×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C24, C4×S3, D12, C22×S3, C4×D4, C23×C4, C22×D4, C2×C4○D4, S3×C2×C4, C2×D12, C4○D12, S3×C23, C2×C4×D4, C4×D12, S3×C22×C4, C22×D12, C2×C4○D12, C2×C4×D12

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

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

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

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

72 conjugacy classes

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

Matrix representation of C2×C4×D12 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 5 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 12 0 0 0 0 12 0 0 0 0 3 10 0 0 3 6
,
 12 0 0 0 0 12 0 0 0 0 3 10 0 0 7 10
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,3,3,0,0,10,6],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,10,10] >;

C2×C4×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC4xD12");
// GroupNames label

G:=SmallGroup(192,1032);
// by ID

G=gap.SmallGroup(192,1032);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,80,6278]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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