Copied to
clipboard

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⋊Q161S3, (S3×Dic6)⋊6C2, (S3×C6).13D4, (C4×S3).10D6, C6.157(S3×D4), (C3×Q8).42D6, C37(Q16⋊S3), D6.9(C3⋊D4), D6.Dic37C2, Q83S3.2S3, C327Q162C2, D12.S39C2, C33(Q8.14D6), C12.20(C22×S3), (C3×C12).20C23, (C3×Dic3).37D4, Dic6⋊S312C2, (S3×C12).21C22, (C3×D12).19C22, C3211(C8.C22), Dic3.18(C3⋊D4), (Q8×C32).2C22, C324C8.10C22, (C3×Dic6).16C22, C324Q8.11C22, C4.20(C2×S32), (C3×C3⋊Q16)⋊5C2, C6.53(C2×C3⋊D4), C2.31(S3×C3⋊D4), (C3×C6).135(C2×D4), (C3×C3⋊C8).14C22, (C3×Q83S3).1C2, SmallGroup(288,591)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.11D6
C1C3C32C3×C6C3×C12S3×C12S3×Dic6 — D12.11D6
C32C3×C6C3×C12 — 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 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], C32, Dic3, Dic3 [×4], C12 [×2], C12 [×6], D6, D6, C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3 [×2], C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, Dic6 [×5], C4×S3, C4×S3 [×2], D12, D12, C2×Dic3, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8 [×2], C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, C4.Dic3, D4.S3 [×2], Q82S3, C3⋊Q16, C3⋊Q16 [×3], C3×Q16, C2×Dic6, S3×Q8, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C324Q8, Q8×C32, Q16⋊S3, Q8.14D6, D6.Dic3, Dic6⋊S3, D12.S3, C3×C3⋊Q16, C327Q16, S3×Dic6, C3×Q83S3, D12.11D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q16⋊S3, Q8.14D6, S3×C3⋊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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 49)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 85)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 74)(38 73)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)
(1 51 5 59 9 55)(2 56 6 52 10 60)(3 49 7 57 11 53)(4 54 8 50 12 58)(13 65 17 61 21 69)(14 70 18 66 22 62)(15 63 19 71 23 67)(16 68 20 64 24 72)(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 82 7 76)(2 81 8 75)(3 80 9 74)(4 79 10 73)(5 78 11 84)(6 77 12 83)(13 93 19 87)(14 92 20 86)(15 91 21 85)(16 90 22 96)(17 89 23 95)(18 88 24 94)(25 52 31 58)(26 51 32 57)(27 50 33 56)(28 49 34 55)(29 60 35 54)(30 59 36 53)(37 72 43 66)(38 71 44 65)(39 70 45 64)(40 69 46 63)(41 68 47 62)(42 67 48 61)

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H12I12J12K24A24B
order1222333444446666668812121212121212121212122424
size11612224246123622412121212364444668888241212

33 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++-+++--
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3⋊D4C3⋊D4C8.C22S32S3×D4C2×S32Q16⋊S3Q8.14D6S3×C3⋊D4D12.11D6
kernelD12.11D6D6.Dic3Dic6⋊S3D12.S3C3×C3⋊Q16C327Q16S3×Dic6C3×Q83S3C3⋊Q16Q83S3C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# 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

׿
×
𝔽