Copied to
clipboard

## G = C8⋊5D12order 192 = 26·3

### 2nd semidirect product of C8 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8⋊5D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C8⋊5D12
 Lower central C3 — C6 — C2×C12 — C8⋊5D12
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C85D12
G = < a,b,c | a8=b12=c2=1, ab=ba, cac=a3, cbc=b-1 >

Subgroups: 536 in 142 conjugacy classes, 55 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C41D4, C4⋊Q8, C2×SD16, C24⋊C2, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C85D4, C4×C24, C122Q8, C4⋊D12, C2×C24⋊C2, C85D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C41D4, C2×SD16, C24⋊C2, C2×D12, C85D4, C4⋊D12, C2×C24⋊C2, C85D12

Smallest permutation representation of C85D12
On 96 points
Generators in S96
(1 61 78 60 20 96 25 37)(2 62 79 49 21 85 26 38)(3 63 80 50 22 86 27 39)(4 64 81 51 23 87 28 40)(5 65 82 52 24 88 29 41)(6 66 83 53 13 89 30 42)(7 67 84 54 14 90 31 43)(8 68 73 55 15 91 32 44)(9 69 74 56 16 92 33 45)(10 70 75 57 17 93 34 46)(11 71 76 58 18 94 35 47)(12 72 77 59 19 95 36 48)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 77)(26 76)(27 75)(28 74)(29 73)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 72)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 D12 D12 C24⋊C2 kernel C8⋊5D12 C4×C24 C12⋊2Q8 C4⋊D12 C2×C24⋊C2 C4×C8 C24 C2×C12 C42 C2×C8 C12 C8 C2×C4 C4 # reps 1 1 1 1 4 1 4 2 1 2 8 8 4 16

Matrix representation of C85D12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 61 0 0 0 0 6 12 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 2 0 0 0 0 72 1 0 0 0 0 0 0 1 2 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 72 2 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,6,0,0,0,0,61,12,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,72,0,0,0,0,2,1,0,0,0,0,0,0,1,72,0,0,0,0,2,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;

C85D12 in GAP, Magma, Sage, TeX

C_8\rtimes_5D_{12}
% in TeX

G:=Group("C8:5D12");
// GroupNames label

G:=SmallGroup(192,252);
// by ID

G=gap.SmallGroup(192,252);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,58,1123,136,6278]);
// Polycyclic

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

׿
×
𝔽