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G = D12.11D6order 288 = 25·32

11st non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.11D6, Dic6.21D6, C3:C8.7D6, Q8.9S32, C3:Q16:1S3, (S3xDic6):6C2, (S3xC6).13D4, (C4xS3).10D6, C6.157(S3xD4), (C3xQ8).42D6, C3:7(Q16:S3), D6.9(C3:D4), D6.Dic3:7C2, Q8:3S3.2S3, C32:7Q16:2C2, D12.S3:9C2, C3:3(Q8.14D6), C12.20(C22xS3), (C3xC12).20C23, (C3xDic3).37D4, Dic6:S3:12C2, (S3xC12).21C22, (C3xD12).19C22, C32:11(C8.C22), Dic3.18(C3:D4), (Q8xC32).2C22, C32:4C8.10C22, (C3xDic6).16C22, C32:4Q8.11C22, C4.20(C2xS32), (C3xC3:Q16):5C2, C6.53(C2xC3:D4), C2.31(S3xC3:D4), (C3xC6).135(C2xD4), (C3xC3:C8).14C22, (C3xQ8:3S3).1C2, SmallGroup(288,591)

Series: Derived Chief Lower central Upper central

C1C3xC12 — D12.11D6
C1C3C32C3xC6C3xC12S3xC12S3xDic6 — D12.11D6
C32C3xC6C3xC12 — D12.11D6
C1C2C4Q8

Generators and relations for D12.11D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=dad-1=a-1, cac-1=a5, cbc-1=a10b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 466 in 129 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3xC6, C3:C8, C3:C8, C24, Dic6, Dic6, C4xS3, C4xS3, D12, D12, C2xDic3, C2xC12, C3xD4, C3xQ8, C3xQ8, C8.C22, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, S3xC6, C8:S3, C24:C2, C4.Dic3, D4.S3, Q8:2S3, C3:Q16, C3:Q16, C3xQ16, C2xDic6, S3xQ8, Q8:3S3, C3xC4oD4, C3xC3:C8, C32:4C8, S3xDic3, C32:2Q8, C3xDic6, S3xC12, S3xC12, C3xD12, C3xD12, C32:4Q8, Q8xC32, Q16:S3, Q8.14D6, D6.Dic3, Dic6:S3, D12.S3, C3xC3:Q16, C32:7Q16, S3xDic6, C3xQ8:3S3, D12.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C8.C22, S32, S3xD4, C2xC3:D4, C2xS32, Q16:S3, Q8.14D6, S3xC3:D4, D12.11D6

Smallest permutation representation of D12.11D6
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 65)(2 64)(3 63)(4 62)(5 61)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 60)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 77)(38 76)(39 75)(40 74)(41 73)(42 84)(43 83)(44 82)(45 81)(46 80)(47 79)(48 78)
(1 49 5 57 9 53)(2 54 6 50 10 58)(3 59 7 55 11 51)(4 52 8 60 12 56)(13 61 17 69 21 65)(14 66 18 62 22 70)(15 71 19 67 23 63)(16 64 20 72 24 68)(25 83 33 75 29 79)(26 76 34 80 30 84)(27 81 35 73 31 77)(28 74 36 78 32 82)(37 88 45 92 41 96)(38 93 46 85 42 89)(39 86 47 90 43 94)(40 91 48 95 44 87)
(1 76 7 82)(2 75 8 81)(3 74 9 80)(4 73 10 79)(5 84 11 78)(6 83 12 77)(13 90 19 96)(14 89 20 95)(15 88 21 94)(16 87 22 93)(17 86 23 92)(18 85 24 91)(25 56 31 50)(26 55 32 49)(27 54 33 60)(28 53 34 59)(29 52 35 58)(30 51 36 57)(37 65 43 71)(38 64 44 70)(39 63 45 69)(40 62 46 68)(41 61 47 67)(42 72 48 66)

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H12I12J12K24A24B
order1222333444446666668812121212121212121212122424
size11612224246123622412121212364444668888241212

33 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++-+++--
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3:D4C3:D4C8.C22S32S3xD4C2xS32Q16:S3Q8.14D6S3xC3:D4D12.11D6
kernelD12.11D6D6.Dic3Dic6:S3D12.S3C3xC3:Q16C32:7Q16S3xDic6C3xQ8:3S3C3:Q16Q8:3S3C3xDic3S3xC6C3:C8Dic6C4xS3D12C3xQ8Dic3D6C32Q8C6C4C3C3C2C1
# reps111111111111111122211112221

Matrix representation of D12.11D6 in GL8(F73)

721000000
720000000
007200000
000720000
00000100
000072000
000000072
00000010
,
10000000
172000000
0043130000
0060300000
0000154707
00004758660
00000661547
0000704758
,
072000000
720000000
00010000
0072720000
0000704758
0000075826
00004758660
00005826066
,
01000000
10000000
0072720000
00010000
0000125656
00002721756
00005617272
000056567271

G:=sub<GL(8,GF(73))| [72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,15,47,0,7,0,0,0,0,47,58,66,0,0,0,0,0,0,66,15,47,0,0,0,0,7,0,47,58],[0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,7,0,47,58,0,0,0,0,0,7,58,26,0,0,0,0,47,58,66,0,0,0,0,0,58,26,0,66],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,2,56,56,0,0,0,0,2,72,17,56,0,0,0,0,56,17,2,72,0,0,0,0,56,56,72,71] >;

D12.11D6 in GAP, Magma, Sage, TeX

D_{12}._{11}D_6
% in TeX

G:=Group("D12.11D6");
// GroupNames label

G:=SmallGroup(288,591);
// by ID

G=gap.SmallGroup(288,591);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,100,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^6,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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