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G = C12.81D12order 288 = 25·32

12nd non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.81D12, C12.14Dic6, Dic3:(C3:C8), C6.21(S3xC8), C32:6(C4:C8), (C3xDic3):1C8, C3:1(C12:C8), (C3xC12).17Q8, C3:2(Dic3:C8), (C3xC12).112D4, (C2xC12).297D6, C62.31(C2xC4), C6.1(C4:Dic3), (C4xDic3).7S3, (C6xDic3).5C4, C6.13(C8:S3), (C3xC6).7M4(2), C12.80(C3:D4), C6.1(Dic3:C4), (Dic3xC12).1C2, C4.7(C32:2Q8), C6.3(C4.Dic3), C4.30(C3:D12), (C6xC12).202C22, (C2xDic3).4Dic3, C2.3(D6.Dic3), C22.11(S3xDic3), C2.1(Dic3:Dic3), C6.5(C2xC3:C8), C2.5(S3xC3:C8), (C6xC3:C8).3C2, (C2xC3:C8).9S3, (C2xC4).130S32, (C2xC6).68(C4xS3), (C3xC6).20(C2xC8), (C3xC6).18(C4:C4), (C2xC6).15(C2xDic3), (C2xC32:4C8).14C2, SmallGroup(288,219)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C12.81D12
C1C3C32C3xC6C3xC12C6xC12Dic3xC12 — C12.81D12
C32C3xC6 — C12.81D12
C1C2xC4

Generators and relations for C12.81D12
 G = < a,b,c | a12=b12=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >

Subgroups: 210 in 87 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2xC4, C2xC4, C32, Dic3, Dic3, C12, C12, C2xC6, C2xC6, C42, C2xC8, C3xC6, C3:C8, C24, C2xDic3, C2xC12, C2xC12, C4:C8, C3xDic3, C3xDic3, C3xC12, C62, C2xC3:C8, C2xC3:C8, C4xDic3, C4xC12, C2xC24, C3xC3:C8, C32:4C8, C6xDic3, C6xC12, C12:C8, Dic3:C8, C6xC3:C8, Dic3xC12, C2xC32:4C8, C12.81D12
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Q8, Dic3, D6, C4:C4, C2xC8, M4(2), C3:C8, Dic6, C4xS3, D12, C2xDic3, C3:D4, C4:C8, S32, S3xC8, C8:S3, C2xC3:C8, C4.Dic3, Dic3:C4, C4:Dic3, S3xDic3, C3:D12, C32:2Q8, C12:C8, Dic3:C8, S3xC3:C8, D6.Dic3, Dic3:Dic3, C12.81D12

Smallest permutation representation of C12.81D12
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 91 79 44 5 87 83 40 9 95 75 48)(2 96 80 37 6 92 84 45 10 88 76 41)(3 89 81 42 7 85 73 38 11 93 77 46)(4 94 82 47 8 90 74 43 12 86 78 39)(13 27 70 53 21 31 66 57 17 35 62 49)(14 32 71 58 22 36 67 50 18 28 63 54)(15 25 72 51 23 29 68 55 19 33 64 59)(16 30 61 56 24 34 69 60 20 26 65 52)
(1 50 10 59 7 56 4 53)(2 55 11 52 8 49 5 58)(3 60 12 57 9 54 6 51)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)(25 81 34 78 31 75 28 84)(26 74 35 83 32 80 29 77)(27 79 36 76 33 73 30 82)(61 94 70 91 67 88 64 85)(62 87 71 96 68 93 65 90)(63 92 72 89 69 86 66 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), (1,91,79,44,5,87,83,40,9,95,75,48)(2,96,80,37,6,92,84,45,10,88,76,41)(3,89,81,42,7,85,73,38,11,93,77,46)(4,94,82,47,8,90,74,43,12,86,78,39)(13,27,70,53,21,31,66,57,17,35,62,49)(14,32,71,58,22,36,67,50,18,28,63,54)(15,25,72,51,23,29,68,55,19,33,64,59)(16,30,61,56,24,34,69,60,20,26,65,52), (1,50,10,59,7,56,4,53)(2,55,11,52,8,49,5,58)(3,60,12,57,9,54,6,51)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45)(25,81,34,78,31,75,28,84)(26,74,35,83,32,80,29,77)(27,79,36,76,33,73,30,82)(61,94,70,91,67,88,64,85)(62,87,71,96,68,93,65,90)(63,92,72,89,69,86,66,95)>;

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I8A8B8C8D8E8F8G8H12A···12H12I12J12K12L12M···12T24A···24H
order1222333444444446···66668888888812···121212121212···1224···24
size1111224111166662···24446666181818182···244446···66···6

60 irreducible representations

dim111111222222222222222444444
type+++++++--+-+++--
imageC1C2C2C2C4C8S3S3D4Q8Dic3D6M4(2)C3:C8Dic6D12C3:D4C4xS3S3xC8C8:S3C4.Dic3S32C3:D12C32:2Q8S3xDic3S3xC3:C8D6.Dic3
kernelC12.81D12C6xC3:C8Dic3xC12C2xC32:4C8C6xDic3C3xDic3C2xC3:C8C4xDic3C3xC12C3xC12C2xDic3C2xC12C3xC6Dic3C12C12C12C2xC6C6C6C6C2xC4C4C4C22C2C2
# reps111148111122244222444111122

Matrix representation of C12.81D12 in GL6(F73)

100000
010000
00464600
0027000
000010
000001
,
22480000
34510000
0072000
001100
0000721
0000720
,
22180000
34510000
0063000
00101000
0000720
0000721

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,34,0,0,0,0,48,51,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,34,0,0,0,0,18,51,0,0,0,0,0,0,63,10,0,0,0,0,0,10,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;

C12.81D12 in GAP, Magma, Sage, TeX

C_{12}._{81}D_{12}
% in TeX

G:=Group("C12.81D12");
// GroupNames label

G:=SmallGroup(288,219);
// by ID

G=gap.SmallGroup(288,219);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations

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