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## G = C12⋊4SD16order 192 = 26·3

### 4th semidirect product of C12 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12⋊4SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12⋊2Q8 — C12⋊4SD16
 Lower central C3 — C6 — C2×C12 — C12⋊4SD16
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C124SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a7, cbc=b3 >

Subgroups: 400 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, C41D4, C4⋊Q8, C2×SD16, C2×C3⋊C8, C4⋊Dic3, D4.S3, C4×C12, C2×Dic6, C6×D4, C6×D4, C85D4, C4×C3⋊C8, C122Q8, C2×D4.S3, C3×C41D4, C124SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C41D4, C2×SD16, D4.S3, S3×D4, C2×C3⋊D4, C85D4, C2×D4.S3, C123D4, C124SD16

Smallest permutation representation of C124SD16
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 80 64 51 92 38 32 16)(2 73 65 56 93 43 33 21)(3 78 66 49 94 48 34 14)(4 83 67 54 95 41 35 19)(5 76 68 59 96 46 36 24)(6 81 69 52 85 39 25 17)(7 74 70 57 86 44 26 22)(8 79 71 50 87 37 27 15)(9 84 72 55 88 42 28 20)(10 77 61 60 89 47 29 13)(11 82 62 53 90 40 30 18)(12 75 63 58 91 45 31 23)
(2 8)(4 10)(6 12)(13 41)(14 48)(15 43)(16 38)(17 45)(18 40)(19 47)(20 42)(21 37)(22 44)(23 39)(24 46)(25 63)(26 70)(27 65)(28 72)(29 67)(30 62)(31 69)(32 64)(33 71)(34 66)(35 61)(36 68)(49 78)(50 73)(51 80)(52 75)(53 82)(54 77)(55 84)(56 79)(57 74)(58 81)(59 76)(60 83)(85 91)(87 93)(89 95)

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

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

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

36 conjugacy classes

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

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 D4.S3 S3×D4 kernel C12⋊4SD16 C4×C3⋊C8 C12⋊2Q8 C2×D4.S3 C3×C4⋊1D4 C4⋊1D4 C3⋊C8 C2×C12 C42 C2×D4 C12 C2×C4 C4 C4 # reps 1 1 1 4 1 1 4 2 1 2 8 4 4 2

Matrix representation of C124SD16 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 0 0 65 0 0 0 0 0 9 9
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 72 2 0 0 0 0 72 1 0 0 0 0 0 0 15 4 0 0 0 0 53 58
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 29 72

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,65,9,0,0,0,0,0,9],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,15,53,0,0,0,0,4,58],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,29,0,0,0,0,0,72] >;

C124SD16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C12:4SD16");
// GroupNames label

G:=SmallGroup(192,635);
// by ID

G=gap.SmallGroup(192,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^7,c*b*c=b^3>;
// generators/relations

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