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## G = (C2×C4)⋊3D12order 192 = 26·3

### 2nd semidirect product of C2×C4 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4)⋊3D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C4)⋊3D12
 Lower central C3 — C22×C6 — (C2×C4)⋊3D12
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4)⋊3D12
G = < a,b,c,d | a2=b4=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 808 in 238 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×20], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×13], D4 [×8], C23, C23 [×16], Dic3 [×2], C12 [×5], D6 [×20], C2×C6 [×3], C2×C6 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C24 [×2], D12 [×8], C2×Dic3 [×6], C2×C12 [×4], C2×C12 [×7], C22×S3 [×4], C22×S3 [×12], C22×C6, C2.C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, D6⋊C4 [×8], C3×C4⋊C4 [×2], C2×D12 [×6], C22×Dic3 [×2], C22×C12 [×3], S3×C23 [×2], C23.10D4, C6.C42, C2×D6⋊C4 [×4], C6×C4⋊C4, C22×D12, (C2×C4)⋊3D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×8], C23, D6 [×3], C2×D4 [×4], C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D12, C4○D12, S3×D4 [×2], Q83S3 [×2], C2×C3⋊D4, C23.10D4, D6.D4 [×2], C12⋊D4 [×2], C127D4, C232D6, C12.23D4, (C2×C4)⋊3D12

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

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4F 4G 4H 4I 4J 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 12 12 2 4 ··· 4 12 12 12 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 C3⋊D4 C4○D12 S3×D4 Q8⋊3S3 kernel (C2×C4)⋊3D12 C6.C42 C2×D6⋊C4 C6×C4⋊C4 C22×D12 C2×C4⋊C4 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 4 1 1 1 4 4 3 6 4 4 4 2 2

Matrix representation of (C2×C4)⋊3D12 in GL6(𝔽13)

 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 4 0 0 0 0 9 11 0 0 0 0 0 0 1 11 0 0 0 0 1 12
,
 3 10 0 0 0 0 3 6 0 0 0 0 0 0 3 10 0 0 0 0 3 6 0 0 0 0 0 0 5 3 0 0 0 0 0 8
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,1,0,0,0,0,11,12],[3,3,0,0,0,0,10,6,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,5,0,0,0,0,0,3,8],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

(C2×C4)⋊3D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3D_{12}
% in TeX

G:=Group("(C2xC4):3D12");
// GroupNames label

G:=SmallGroup(192,550);
// by ID

G=gap.SmallGroup(192,550);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,184,6278]);
// 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,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations

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