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G = D6:4D12order 288 = 25·32

1st semidirect product of D6 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6:4D12, C62.92C23, (S3xC6):3D4, D6:C4:18S3, (C2xD12):6S3, (C2xC12):13D6, C6.26(S3xD4), (C6xD12):23C2, D6:5(C3:D4), (C2xDic3):1D6, C6.28(C2xD12), C2.28(S3xD12), C32:2C22wrC2, (C22xS3):2D6, C3:3(D6:D4), C3:1(C23:2D6), D6:Dic3:14C2, (C6xC12):17C22, (C6xDic3):1C22, C6.11D12:12C2, C2.20(D6:D6), (C2xC4):1S32, (C2xC3:S3):2D4, (C22xS32):1C2, (S3xC2xC6):1C22, (C3xD6:C4):15C2, C2.19(S3xC3:D4), C6.41(C2xC3:D4), (C2xD6:S3):6C2, (C2xC3:D12):7C2, C22.127(C2xS32), (C3xC6).114(C2xD4), (C2xC3:Dic3):2C22, (C2xC6).111(C22xS3), (C22xC3:S3).27C22, SmallGroup(288,570)

Series: Derived Chief Lower central Upper central

C1C62 — D6:4D12
C1C3C32C3xC6C62S3xC2xC6C22xS32 — D6:4D12
C32C62 — D6:4D12
C1C22C2xC4

Generators and relations for D6:4D12
 G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c-1 >

Subgroups: 1370 in 277 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C2xD4, C24, C3xS3, C3:S3, C3xC6, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22wrC2, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, D6:C4, D6:C4, C6.D4, C3xC22:C4, C2xD12, C2xD12, C2xC3:D4, C6xD4, S3xC23, D6:S3, C3:D12, C3xD12, C6xDic3, C2xC3:Dic3, C6xC12, C2xS32, S3xC2xC6, C22xC3:S3, D6:D4, C23:2D6, D6:Dic3, C3xD6:C4, C6.11D12, C2xD6:S3, C2xC3:D12, C6xD12, C22xS32, D6:4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C3:D4, C22xS3, C22wrC2, S32, C2xD12, S3xD4, C2xC3:D4, C2xS32, D6:D4, C23:2D6, S3xD12, D6:D6, S3xC3:D4, D6:4D12

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G6H6I6J···6O12A···12H12I12J
order122222222223334446···66666···612···121212
size11116666121818224412362···244412···124···41212

42 irreducible representations

dim11111111222222222444444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D12C3:D4S32S3xD4C2xS32S3xD12D6:D6S3xC3:D4
kernelD6:4D12D6:Dic3C3xD6:C4C6.11D12C2xD6:S3C2xC3:D12C6xD12C22xS32D6:C4C2xD12S3xC6C2xC3:S3C2xDic3C2xC12C22xS3D6D6C2xC4C6C22C2C2C2
# reps11111111114212344141222

Matrix representation of D6:4D12 in GL8(F13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
012000000
120000000
00100000
00010000
000012000
000012100
000000120
000000121
,
80000000
05000000
001120000
00100000
000012000
000001200
000000111
000000012
,
05000000
80000000
00100000
001120000
000012000
000001200
000000111
000000012

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,12],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,12] >;

D6:4D12 in GAP, Magma, Sage, TeX

D_6\rtimes_4D_{12}
% in TeX

G:=Group("D6:4D12");
// GroupNames label

G:=SmallGroup(288,570);
// by ID

G=gap.SmallGroup(288,570);
# by ID

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

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

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