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G = D12.24D6order 288 = 25·32

9th non-split extension by D12 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D12.24D6, Dic6.11D6, C3⋊C8.9D6, (S3×Q8)⋊5S3, Q8.10S32, Q82S34S3, (C4×S3).11D6, (S3×C6).37D4, C6.158(S3×D4), (C3×Q8).33D6, C37(D4.D6), D6.Dic34C2, C327Q163C2, D125S3.2C2, D6.15(C3⋊D4), (C3×C12).23C23, C12.23(C22×S3), (C3×Dic3).17D4, C323Q1613C2, Dic6⋊S313C2, C32(Q8.11D6), (S3×C12).22C22, (C3×D12).20C22, C3213(C8.C22), Dic3.12(C3⋊D4), (Q8×C32).5C22, C324C8.11C22, (C3×Dic6).19C22, C324Q8.13C22, (C3×S3×Q8)⋊2C2, C4.23(C2×S32), C6.54(C2×C3⋊D4), C2.32(S3×C3⋊D4), (C3×Q82S3)⋊3C2, (C3×C6).138(C2×D4), (C3×C3⋊C8).12C22, SmallGroup(288,594)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.24D6
C1C3C32C3×C6C3×C12S3×C12D125S3 — D12.24D6
C32C3×C6C3×C12 — D12.24D6
C1C2C4Q8

Generators and relations for D12.24D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=c5 >

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 [×4], C4×S3, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C3×D4, C3×Q8 [×2], C3×Q8 [×3], C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C8⋊S3, Dic12, C4.Dic3, D4.S3, Q82S3, Q82S3, C3⋊Q16 [×4], C3×SD16, C4○D12, D42S3, S3×Q8, C6×Q8, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C324Q8, Q8×C32, D4.D6, Q8.11D6, D6.Dic3, Dic6⋊S3, C323Q16, C3×Q82S3, C327Q16, D125S3, C3×S3×Q8, D12.24D6
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, D4.D6, Q8.11D6, S3×C3⋊D4, D12.24D6

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

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

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

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

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B12A12B12C12D12E12F12G12H12I12J12K24A24B
order1222333444446666668812121212121212121212122424
size11612224246123622466241236444488881212121212

33 irreducible representations

dim111111112222222222244444448
type+++++++++++++++++-+++--
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3⋊D4C3⋊D4C8.C22S32S3×D4C2×S32D4.D6Q8.11D6S3×C3⋊D4D12.24D6
kernelD12.24D6D6.Dic3Dic6⋊S3C323Q16C3×Q82S3C327Q16D125S3C3×S3×Q8Q82S3S3×Q8C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps111111111111111122211112221

Matrix representation of D12.24D6 in GL8(𝔽73)

072000000
11000000
00110000
007200000
00000100
000072000
000013297263
00002513441
,
00110000
007200000
072000000
11000000
00001967608
0000363508
000030661771
00004920222
,
4330000000
4313000000
0030430000
0030600000
000010660
00001856663
0000420720
00004825017
,
0034260000
0065390000
3947000000
834000000
000038572054
00005367530
000041585919
000040616955

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,13,25,0,0,0,0,1,0,29,13,0,0,0,0,0,0,72,44,0,0,0,0,0,0,63,1],[0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,19,36,30,49,0,0,0,0,67,35,66,20,0,0,0,0,60,0,17,22,0,0,0,0,8,8,71,2],[43,43,0,0,0,0,0,0,30,13,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,18,42,48,0,0,0,0,0,56,0,25,0,0,0,0,66,66,72,0,0,0,0,0,0,3,0,17],[0,0,39,8,0,0,0,0,0,0,47,34,0,0,0,0,34,65,0,0,0,0,0,0,26,39,0,0,0,0,0,0,0,0,0,0,38,53,41,40,0,0,0,0,57,67,58,61,0,0,0,0,20,53,59,69,0,0,0,0,54,0,19,55] >;

D12.24D6 in GAP, Magma, Sage, TeX

D_{12}._{24}D_6
% in TeX

G:=Group("D12.24D6");
// GroupNames label

G:=SmallGroup(288,594);
// by ID

G=gap.SmallGroup(288,594);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,100,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=c^5>;
// generators/relations

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