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G = D12.12D6order 288 = 25·32

12nd non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.12D6, C3⋊C8.17D6, Q8.11S32, Q82S37S3, Q83S35S3, (S3×C6).14D4, (C4×S3).24D6, C6.159(S3×D4), C322D89C2, (C3×Q8).43D6, D125S36C2, D6.3(C3⋊D4), C3213(C4○D8), C327Q164C2, C34(Q8.13D6), C37(Q8.7D6), (C3×C12).24C23, C12.24(C22×S3), (C3×Dic3).41D4, D12.S314C2, (S3×C12).23C22, (C3×D12).21C22, Dic3.22(C3⋊D4), (Q8×C32).6C22, C324C8.12C22, C324Q8.14C22, (S3×C3⋊C8)⋊7C2, C4.24(C2×S32), C6.55(C2×C3⋊D4), C2.33(S3×C3⋊D4), (C3×Q83S3)⋊2C2, (C3×Q82S3)⋊7C2, (C3×C6).139(C2×D4), (C3×C3⋊C8).19C22, SmallGroup(288,595)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.12D6
C1C3C32C3×C6C3×C12S3×C12D125S3 — D12.12D6
C32C3×C6C3×C12 — D12.12D6
C1C2C4Q8

Generators and relations for D12.12D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=dad-1=a-1, cac-1=a5, cbc-1=a10b, dbd-1=ab, dcd-1=a6c-1 >

Subgroups: 498 in 133 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×3], C6 [×2], C6 [×4], C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, C32, Dic3, Dic3 [×3], C12 [×2], C12 [×5], D6, D6 [×2], C2×C6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×3], C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6 [×3], C4×S3, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4 [×3], C3×Q8 [×2], C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6 [×2], S3×C8, C24⋊C2, C2×C3⋊C8, D4⋊S3 [×2], D4.S3, Q82S3, C3⋊Q16 [×3], C3×SD16, C4○D12, D42S3, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, S3×C12, S3×C12, C3×D12 [×2], C3×D12, C324Q8, Q8×C32, Q8.7D6, Q8.13D6, S3×C3⋊C8, C322D8, D12.S3, C3×Q82S3, C327Q16, D125S3, C3×Q83S3, D12.12D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8.7D6, Q8.13D6, S3×C3⋊D4, D12.12D6

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

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

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

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

36 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F12G12H12I12J24A24B
order122223334444466666668888121212121212121212122424
size11612122242334362241212122466181844446688881212

36 irreducible representations

dim11111111222222222224444448
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C4○D8S32S3×D4C2×S32Q8.7D6Q8.13D6S3×C3⋊D4D12.12D6
kernelD12.12D6S3×C3⋊C8C322D8D12.S3C3×Q82S3C327Q16D125S3C3×Q83S3Q82S3Q83S3C3×Dic3S3×C6C3⋊C8C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps11111111111111222241112221

Matrix representation of D12.12D6 in GL6(𝔽73)

1480000
3720000
0072100
0072000
000010
000001
,
52160000
9210000
001000
0017200
000010
000001
,
27550000
8460000
000100
001000
0000172
000010
,
37620000
25360000
000100
001000
0000720
0000721

G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,9,0,0,0,0,16,21,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,8,0,0,0,0,55,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[37,25,0,0,0,0,62,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;

D12.12D6 in GAP, Magma, Sage, TeX

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

G:=Group("D12.12D6");
// GroupNames label

G:=SmallGroup(288,595);
// by ID

G=gap.SmallGroup(288,595);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,1356,9414]);
// 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*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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