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G = D12.33D6order 288 = 25·32

8th non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.33D6, Dic6.34D6, C3212- 1+4, C62.129C23, C4○D125S3, C3⋊D4.3D6, C31(Q8○D12), (S3×Dic6)⋊7C2, (C4×S3).13D6, C6.4(S3×C23), (C3×C6).4C24, D12⋊S37C2, (C2×Dic6)⋊13S3, (C6×Dic6)⋊19C2, D6.6D67C2, D6.3D61C2, (C2×C12).166D6, (S3×C6).2C23, C3⋊D12.C22, D6.3(C22×S3), C12.59D67C2, (C2×Dic3).49D6, Dic3.D610C2, (S3×C12).30C22, (C6×C12).159C22, (C3×C12).113C23, C12.130(C22×S3), C31(Q8.15D6), (C3×D12).42C22, C3⋊Dic3.15C23, (S3×Dic3).1C22, (C3×Dic3).3C23, Dic3.2(C22×S3), C322Q8.4C22, C6.D6.5C22, C327D4.2C22, C12⋊S3.33C22, (C3×Dic6).43C22, (C6×Dic3).45C22, C324Q8.34C22, C4.61(C2×S32), (C2×C4).35S32, C22.6(C2×S32), C2.7(C22×S32), (C3×C4○D12)⋊9C2, (C4×C3⋊S3).41C22, (C2×C3⋊S3).17C23, (C2×C6).12(C22×S3), (C3×C3⋊D4).2C22, SmallGroup(288,945)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D12.33D6
C1C3C32C3×C6S3×C6S3×Dic3S3×Dic6 — D12.33D6
C32C3×C6 — D12.33D6
C1C2C2×C4

Generators and relations for D12.33D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a6c5 >

Subgroups: 1050 in 312 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×2], C4 [×8], C22, C22 [×4], S3 [×8], C6 [×2], C6 [×6], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], C32, Dic3 [×6], Dic3 [×6], C12 [×4], C12 [×8], D6 [×2], D6 [×6], C2×C6 [×2], C2×C6 [×3], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6, Dic6, Dic6 [×4], Dic6 [×11], C4×S3 [×2], C4×S3 [×18], D12, D12 [×7], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×5], C3×D4 [×3], C3×Q8 [×5], 2- 1+4, C3×Dic3 [×6], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C62, C2×Dic6, C2×Dic6 [×2], C4○D12, C4○D12 [×9], D42S3 [×6], S3×Q8 [×6], Q83S3 [×4], C6×Q8, C3×C4○D4, S3×Dic3 [×4], C6.D6 [×4], C3⋊D12 [×4], C322Q8 [×4], C3×Dic6, C3×Dic6 [×4], S3×C12 [×2], C3×D12, C6×Dic3 [×2], C3×C3⋊D4 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, Q8.15D6, Q8○D12, S3×Dic6 [×2], D12⋊S3 [×2], Dic3.D6 [×2], D6.6D6 [×2], D6.3D6 [×4], C6×Dic6, C3×C4○D12, C12.59D6, D12.33D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8.15D6, Q8○D12, C22×S32, D12.33D6

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

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C···4H4I4J6A6B6C6D6E6F6G6H6I6J12A12B12C···12I12J···12O
order1222222333444···4446666666666121212···1212···12
size112661818224226···61818222244441212224···412···12

45 irreducible representations

dim111111111222222224444444
type+++++++++++++++++-+++-
imageC1C2C2C2C2C2C2C2C2S3S3D6D6D6D6D6D62- 1+4S32C2×S32C2×S32Q8.15D6Q8○D12D12.33D6
kernelD12.33D6S3×Dic6D12⋊S3Dic3.D6D6.6D6D6.3D6C6×Dic6C3×C4○D12C12.59D6C2×Dic6C4○D12Dic6C4×S3D12C2×Dic3C3⋊D4C2×C12C32C2×C4C4C22C3C3C1
# reps122224111115212221121224

Matrix representation of D12.33D6 in GL4(𝔽13) generated by

8800
5000
0055
0080
,
001212
0001
121200
0100
,
71000
31000
00103
00107
,
00310
0036
71000
31000
G:=sub<GL(4,GF(13))| [8,5,0,0,8,0,0,0,0,0,5,8,0,0,5,0],[0,0,12,0,0,0,12,1,12,0,0,0,12,1,0,0],[7,3,0,0,10,10,0,0,0,0,10,10,0,0,3,7],[0,0,7,3,0,0,10,10,3,3,0,0,10,6,0,0] >;

D12.33D6 in GAP, Magma, Sage, TeX

D_{12}._{33}D_6
% in TeX

G:=Group("D12.33D6");
// GroupNames label

G:=SmallGroup(288,945);
// by ID

G=gap.SmallGroup(288,945);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,1356,9414]);
// Polycyclic

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

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