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G = D12.13D6order 288 = 25·32

13rd non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.13D6, Dic6.23D6, C3⋊C8.18D6, Q8.13S32, C3⋊Q167S3, C3⋊D249C2, Q83S36S3, (S3×C6).15D4, (C4×S3).25D6, C6.161(S3×D4), (C3×Q8).45D6, D6.6D66C2, D6.4(C3⋊D4), C3214(C4○D8), C35(D24⋊C2), C35(Q8.13D6), C3211SD164C2, C12.26(C22×S3), (C3×C12).26C23, (C3×Dic3).42D4, Dic6⋊S314C2, (S3×C12).25C22, (C3×D12).22C22, C12⋊S3.14C22, Dic3.23(C3⋊D4), (Q8×C32).8C22, C324C8.14C22, (C3×Dic6).21C22, (S3×C3⋊C8)⋊3C2, C4.26(C2×S32), (C3×C3⋊Q16)⋊3C2, C6.57(C2×C3⋊D4), C2.35(S3×C3⋊D4), (C3×C3⋊C8).8C22, (C3×Q83S3)⋊3C2, (C3×C6).141(C2×D4), SmallGroup(288,597)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.13D6
C1C3C32C3×C6C3×C12S3×C12D6.6D6 — D12.13D6
C32C3×C6C3×C12 — D12.13D6
C1C2C4Q8

Generators and relations for D12.13D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=ab, dcd=a6c-1 >

Subgroups: 554 in 134 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×5], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6 [×4], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, C4×S3, C4×S3 [×3], D12, D12 [×5], C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, S3×C8, D24, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3 [×4], C3⋊Q16, C3×Q16, C4○D12, Q83S3, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, C6.D6, C3⋊D12, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C12⋊S3, Q8×C32, D24⋊C2, Q8.13D6, S3×C3⋊C8, C3⋊D24, Dic6⋊S3, C3×C3⋊Q16, C3211SD16, D6.6D6, C3×Q83S3, D12.13D6
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, D24⋊C2, Q8.13D6, S3×C3⋊D4, D12.13D6

Smallest permutation representation of D12.13D6
On 48 points
Generators in S48
(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)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 26 5 34 9 30)(2 31 6 27 10 35)(3 36 7 32 11 28)(4 29 8 25 12 33)(13 38 17 46 21 42)(14 43 18 39 22 47)(15 48 19 44 23 40)(16 41 20 37 24 45)
(1 11)(2 10)(3 9)(4 8)(5 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 27)(28 36)(29 35)(30 34)(31 33)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)

G:=sub<Sym(48)| (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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,26,5,34,9,30)(2,31,6,27,10,35)(3,36,7,32,11,28)(4,29,8,25,12,33)(13,38,17,46,21,42)(14,43,18,39,22,47)(15,48,19,44,23,40)(16,41,20,37,24,45), (1,11)(2,10)(3,9)(4,8)(5,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,27)(28,36)(29,35)(30,34)(31,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)>;

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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,26,5,34,9,30)(2,31,6,27,10,35)(3,36,7,32,11,28)(4,29,8,25,12,33)(13,38,17,46,21,42)(14,43,18,39,22,47)(15,48,19,44,23,40)(16,41,20,37,24,45), (1,11)(2,10)(3,9)(4,8)(5,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,27)(28,36)(29,35)(30,34)(31,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43) );

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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)], [(1,26,5,34,9,30),(2,31,6,27,10,35),(3,36,7,32,11,28),(4,29,8,25,12,33),(13,38,17,46,21,42),(14,43,18,39,22,47),(15,48,19,44,23,40),(16,41,20,37,24,45)], [(1,11),(2,10),(3,9),(4,8),(5,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,27),(28,36),(29,35),(30,34),(31,33),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)])

36 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H12I12J12K24A24B
order1222233344444666666888812121212121212121212122424
size11612362242334122241212126618184444668888241212

36 irreducible representations

dim111111112222222222224444448
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6C3⋊D4C3⋊D4C4○D8S32S3×D4C2×S32D24⋊C2Q8.13D6S3×C3⋊D4D12.13D6
kernelD12.13D6S3×C3⋊C8C3⋊D24Dic6⋊S3C3×C3⋊Q16C3211SD16D6.6D6C3×Q83S3C3⋊Q16Q83S3C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps111111111111111122241112221

Matrix representation of D12.13D6 in GL6(𝔽73)

7230000
4810000
0072100
0072000
000010
000001
,
32250000
35410000
0072000
0072100
0000720
0000072
,
4680000
55270000
0007200
0072000
0000072
000011
,
7200000
4810000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [72,48,0,0,0,0,3,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,35,0,0,0,0,25,41,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[46,55,0,0,0,0,8,27,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[72,48,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D12.13D6 in GAP, Magma, Sage, TeX

D_{12}._{13}D_6
% in TeX

G:=Group("D12.13D6");
// GroupNames label

G:=SmallGroup(288,597);
// by ID

G=gap.SmallGroup(288,597);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,422,135,100,346,185,80,1356,9414]);
// Polycyclic

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

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