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G = C12⋊6SD16order 192 = 26·3

6th semidirect product of C12 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12⋊6SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C12⋊6SD16
 Lower central C3 — C6 — C2×C12 — C12⋊6SD16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C126SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=b3 >

Subgroups: 496 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×6], C4 [×2], C22, C22 [×6], S3 [×2], C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], C12 [×6], C12 [×2], D6 [×6], C2×C6, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×8], C2×D4 [×4], C2×Q8 [×2], C3⋊C8 [×4], D12 [×8], C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×4], C22×S3 [×2], C4×C8, C41D4, C4⋊Q8, C2×SD16 [×4], C2×C3⋊C8 [×2], Q82S3 [×8], C4×C12, C3×C4⋊C4 [×2], C2×D12 [×2], C2×D12 [×2], C6×Q8 [×2], C85D4, C4×C3⋊C8, C4⋊D12, C2×Q82S3 [×4], C3×C4⋊Q8, C126SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×4], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C2×SD16 [×2], Q82S3 [×4], S3×D4 [×2], C2×C3⋊D4, C85D4, C123D4, C2×Q82S3 [×2], C126SD16

Smallest permutation representation of C126SD16
On 96 points
Generators in S96
(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 49 45 73 14 88 34 68)(2 54 46 78 15 93 35 61)(3 59 47 83 16 86 36 66)(4 52 48 76 17 91 25 71)(5 57 37 81 18 96 26 64)(6 50 38 74 19 89 27 69)(7 55 39 79 20 94 28 62)(8 60 40 84 21 87 29 67)(9 53 41 77 22 92 30 72)(10 58 42 82 23 85 31 65)(11 51 43 75 24 90 32 70)(12 56 44 80 13 95 33 63)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 15)(16 24)(17 23)(18 22)(19 21)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(49 73)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 96)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4F 4G 4H 6A 6B 6C 8A ··· 8H 12A ··· 12F 12G 12H 12I 12J order 1 2 2 2 2 2 3 4 ··· 4 4 4 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 size 1 1 1 1 24 24 2 2 ··· 2 8 8 2 2 2 6 ··· 6 4 ··· 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 C3⋊D4 Q8⋊2S3 S3×D4 kernel C12⋊6SD16 C4×C3⋊C8 C4⋊D12 C2×Q8⋊2S3 C3×C4⋊Q8 C4⋊Q8 C3⋊C8 C2×C12 C42 C2×Q8 C12 C2×C4 C4 C4 # reps 1 1 1 4 1 1 4 2 1 2 8 4 4 2

Matrix representation of C126SD16 in GL6(𝔽73)

 0 72 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 67 67 0 0 0 0 6 67 0 0 0 0 0 0 6 6 0 0 0 0 67 6
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,67,6,0,0,0,0,67,67,0,0,0,0,0,0,6,67,0,0,0,0,6,6],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C126SD16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("C12:6SD16");
// GroupNames label

G:=SmallGroup(192,644);
// by ID

G=gap.SmallGroup(192,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

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