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G = C89D12order 192 = 26·3

3rd semidirect product of C8 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C89D12, C2419D4, D61M4(2), C42.15D6, C8⋊C48S3, D6⋊C836C2, C31(C89D4), D6⋊C4.4C4, C12⋊C83C2, C6.10(C4×D4), (C4×D12).4C2, (C2×D12).7C4, C2.13(C4×D12), (C2×C8).157D6, C4.77(C2×D12), C4⋊Dic3.9C4, C6.21(C8○D4), C12.297(C2×D4), C2.7(D12.C4), (C4×C12).13C22, C6.17(C2×M4(2)), C2.10(S3×M4(2)), C4.130(C4○D12), C12.246(C4○D4), (C2×C24).269C22, (C2×C12).812C23, (S3×C2×C8)⋊26C2, (C2×C4).29(C4×S3), (C3×C8⋊C4)⋊12C2, (C2×C8⋊S3)⋊24C2, C22.99(S3×C2×C4), (C2×C12).37(C2×C4), (C2×C3⋊C8).295C22, (S3×C2×C4).177C22, (C2×C6).67(C22×C4), (C22×S3).33(C2×C4), (C2×C4).754(C22×S3), (C2×Dic3).14(C2×C4), SmallGroup(192,265)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C89D12
C1C3C6C12C2×C12S3×C2×C4C4×D12 — C89D12
C3C2×C6 — C89D12
C1C2×C4C8⋊C4

Generators and relations for C89D12
 G = < a,b,c | a8=b12=c2=1, bab-1=cac=a5, cbc=b-1 >

Subgroups: 312 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C24, C24, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), S3×C8, C8⋊S3, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C89D4, C12⋊C8, D6⋊C8, C3×C8⋊C4, C4×D12, S3×C2×C8, C2×C8⋊S3, C89D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, C2×D12, C4○D12, C89D4, C4×D12, S3×M4(2), D12.C4, C89D12

Smallest permutation representation of C89D12
On 96 points
Generators in S96
(1 16 56 81 71 34 43 86)(2 35 57 87 72 17 44 82)(3 18 58 83 61 36 45 88)(4 25 59 89 62 19 46 84)(5 20 60 73 63 26 47 90)(6 27 49 91 64 21 48 74)(7 22 50 75 65 28 37 92)(8 29 51 93 66 23 38 76)(9 24 52 77 67 30 39 94)(10 31 53 95 68 13 40 78)(11 14 54 79 69 32 41 96)(12 33 55 85 70 15 42 80)
(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 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(37 39)(40 48)(41 47)(42 46)(43 45)(49 53)(50 52)(54 60)(55 59)(56 58)(61 71)(62 70)(63 69)(64 68)(65 67)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D8E8F8G8H8I8J8K8L12A12B12C12D12E12F12G12H24A···24H
order12222223444444444666888888888888121212121212121224···24
size111166122111144661222222224466661212222244444···4

48 irreducible representations

dim1111111111222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C4C4C4S3D4D6D6C4○D4M4(2)D12C4×S3C8○D4C4○D12S3×M4(2)D12.C4
kernelC89D12C12⋊C8D6⋊C8C3×C8⋊C4C4×D12S3×C2×C8C2×C8⋊S3C4⋊Dic3D6⋊C4C2×D12C8⋊C4C24C42C2×C8C12D6C8C2×C4C6C4C2C2
# reps1121111242121224444422

Matrix representation of C89D12 in GL6(𝔽73)

7200000
0720000
001000
000100
00002671
00002347
,
60670000
4130000
001100
0072000
000010
00002672
,
7200000
5310000
00727200
000100
000010
00002672

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,23,0,0,0,0,71,47],[60,4,0,0,0,0,67,13,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,26,0,0,0,0,0,72],[72,53,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,26,0,0,0,0,0,72] >;

C89D12 in GAP, Magma, Sage, TeX

C_8\rtimes_9D_{12}
% in TeX

G:=Group("C8:9D12");
// GroupNames label

G:=SmallGroup(192,265);
// by ID

G=gap.SmallGroup(192,265);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,136,6278]);
// Polycyclic

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

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