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## G = (C2×C4)⋊6D12order 192 = 26·3

### 1st semidirect product of C2×C4 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C4)⋊6D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C4)⋊6D12
 Lower central C3 — C2×C6 — (C2×C4)⋊6D12
 Upper central C1 — C23 — C2×C42

Generators and relations for (C2×C4)⋊6D12
G = < a,b,c,d | a2=b4=c12=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=c-1 >

Subgroups: 824 in 258 conjugacy classes, 87 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×10], D4 [×8], C23, C23 [×16], Dic3 [×2], C12 [×4], C12 [×4], D6 [×20], C2×C6 [×3], C2×C6 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C24 [×2], D12 [×8], C2×Dic3 [×6], C2×C12 [×10], C2×C12 [×4], C22×S3 [×4], C22×S3 [×12], C22×C6, C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C4⋊Dic3 [×2], D6⋊C4 [×8], C4×C12 [×2], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23 [×2], C24.3C22, C2×C4⋊Dic3, C2×D6⋊C4 [×4], C2×C4×C12, C22×D12, (C2×C4)⋊6D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], D12 [×6], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, D6⋊C4 [×4], S3×C2×C4, C2×D12 [×3], C4○D12 [×2], C2×C3⋊D4, C24.3C22, C4×D12 [×2], C4⋊D12, C427S3, C2×D6⋊C4, C127D4 [×2], (C2×C4)⋊6D12

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4L 4M 4N 4O 4P 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 12 12 12 12 2 2 ··· 2 12 12 12 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 C4○D4 C4×S3 D12 C3⋊D4 C4○D12 kernel (C2×C4)⋊6D12 C2×C4⋊Dic3 C2×D6⋊C4 C2×C4×C12 C22×D12 C2×D12 C2×C42 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 # reps 1 1 4 1 1 8 1 8 3 4 4 12 4 8

Matrix representation of (C2×C4)⋊6D12 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 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 3 0 0 0 0 5 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 0 0 0 7 3 0 0 0 0 10 10
,
 12 0 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 12 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,5,0,0,0,0,3,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;

(C2×C4)⋊6D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_6D_{12}
% in TeX

G:=Group("(C2xC4):6D12");
// GroupNames label

G:=SmallGroup(192,498);
// by ID

G=gap.SmallGroup(192,498);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,758,58,6278]);
// Polycyclic

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

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