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G = C12.77D12order 288 = 25·32

8th non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.77D12, D6:(C3:C8), (S3xC6):1C8, C3:3(D6:C8), C6.20(S3xC8), C6.31(D6:C4), (C2xC12).295D6, (C3xC12).107D4, C32:4(C22:C8), C62.25(C2xC4), (C6xDic3).4C4, C6.12(C8:S3), (C3xC6).5M4(2), C12.94(C3:D4), C2.1(D6:Dic3), C6.2(C4.Dic3), C4.16(D6:S3), C4.26(C3:D12), C3:1(C12.55D4), (C6xC12).200C22, (C2xDic3).3Dic3, C6.1(C6.D4), C2.2(D6.Dic3), (C22xS3).2Dic3, C22.10(S3xDic3), (C6xC3:C8):1C2, (C2xC3:C8):9S3, C6.4(C2xC3:C8), C2.4(S3xC3:C8), (S3xC2xC4).7S3, (S3xC2xC6).3C4, (C2xC4).128S32, (S3xC2xC12).16C2, (C2xC6).65(C4xS3), (C3xC6).18(C2xC8), (C2xC32:4C8):14C2, (C2xC6).14(C2xDic3), (C3xC6).23(C22:C4), SmallGroup(288,204)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C12.77D12
C1C3C32C3xC6C3xC12C6xC12S3xC2xC12 — C12.77D12
C32C3xC6 — C12.77D12
C1C2xC4

Generators and relations for C12.77D12
 G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b11 >

Subgroups: 306 in 111 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, C22xC4, C3xS3, C3xC6, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, C22:C8, C3xDic3, C3xC12, S3xC6, S3xC6, C62, C2xC3:C8, C2xC3:C8, C2xC24, S3xC2xC4, C22xC12, C3xC3:C8, C32:4C8, S3xC12, C6xDic3, C6xC12, S3xC2xC6, D6:C8, C12.55D4, C6xC3:C8, C2xC32:4C8, S3xC2xC12, C12.77D12
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Dic3, D6, C22:C4, C2xC8, M4(2), C3:C8, C4xS3, D12, C2xDic3, C3:D4, C22:C8, S32, S3xC8, C8:S3, C2xC3:C8, C4.Dic3, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, D6:C8, C12.55D4, S3xC3:C8, D6.Dic3, D6:Dic3, C12.77D12

Smallest permutation representation of C12.77D12
On 96 points
Generators in S96
(1 74 21 94 17 90 13 86 9 82 5 78)(2 91 6 95 10 75 14 79 18 83 22 87)(3 76 23 96 19 92 15 88 11 84 7 80)(4 93 8 73 12 77 16 81 20 85 24 89)(25 67 45 63 41 59 37 55 33 51 29 71)(26 60 30 64 34 68 38 72 42 52 46 56)(27 69 47 65 43 61 39 57 35 53 31 49)(28 62 32 66 36 70 40 50 44 54 48 58)
(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 51 82 37 13 63 94 25)(2 36 83 62 14 48 95 50)(3 61 84 47 15 49 96 35)(4 46 85 72 16 34 73 60)(5 71 86 33 17 59 74 45)(6 32 87 58 18 44 75 70)(7 57 88 43 19 69 76 31)(8 42 89 68 20 30 77 56)(9 67 90 29 21 55 78 41)(10 28 91 54 22 40 79 66)(11 53 92 39 23 65 80 27)(12 38 93 64 24 26 81 52)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M8A8B8C8D8E8F8G8H12A···12H12I12J12K12L12M12N12O12P24A···24H
order1222223334444446···666666668888888812···12121212121212121224···24
size1111662241111662···244466666666181818182···2444466666···6

60 irreducible representations

dim111111122222222222222444444
type+++++++-+-++-+-
imageC1C2C2C2C4C4C8S3S3D4Dic3D6Dic3M4(2)D12C3:D4C3:C8C4xS3S3xC8C8:S3C4.Dic3S32D6:S3C3:D12S3xDic3S3xC3:C8D6.Dic3
kernelC12.77D12C6xC3:C8C2xC32:4C8S3xC2xC12C6xDic3S3xC2xC6S3xC6C2xC3:C8S3xC2xC4C3xC12C2xDic3C2xC12C22xS3C3xC6C12C12D6C2xC6C6C6C6C2xC4C4C4C22C2C2
# reps111122811212122642444111122

Matrix representation of C12.77D12 in GL6(F73)

100000
010000
0046000
0004600
0000072
0000172
,
010000
7200000
00512200
0051000
000001
000010
,
30140000
14430000
0005100
0051000
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,22,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[30,14,0,0,0,0,14,43,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C12.77D12 in GAP, Magma, Sage, TeX

C_{12}._{77}D_{12}
% in TeX

G:=Group("C12.77D12");
// GroupNames label

G:=SmallGroup(288,204);
// by ID

G=gap.SmallGroup(288,204);
# by ID

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

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations

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