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G = D12:27D6order 288 = 25·32

3rd semidirect product of D12 and D6 acting through Inn(D12)

metabelian, supersoluble, monomial

Aliases: D12:27D6, Dic6:26D6, C32:22+ 1+4, C62.140C23, (C4xS3):2D6, (C2xC12):6D6, C3:D4:7D6, C4oD12:9S3, (S3xD12):8C2, Dic3:D6:3C2, C3:2(D4oD12), (C6xC12):8C22, (S3xC12):3C22, D6.6D6:8C2, (S3xC6).7C23, C6.15(S3xC23), (C3xC6).15C24, D6.8(C22xS3), (C3xD12):32C22, C3:D12:2C22, C6.D6:2C22, C12:S3:24C22, C12.133(C22xS3), (C3xC12).120C23, (C3xDic6):31C22, Dic3.7(C22xS3), (C3xDic3).10C23, (C2xC4):4S32, C4.64(C2xS32), (C2xS32):3C22, C2.17(C22xS32), C22.13(C2xS32), (C3xC4oD12):14C2, (C2xC12:S3):19C2, (C3xC3:D4):9C22, (C2xC3:S3).21C23, (C22xC3:S3):6C22, (C2xC6).15(C22xS3), SmallGroup(288,956)

Series: Derived Chief Lower central Upper central

C1C3xC6 — D12:27D6
C1C3C32C3xC6S3xC6C2xS32S3xD12 — D12:27D6
C32C3xC6 — D12:27D6
C1C2C2xC4

Generators and relations for D12:27D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a6b, dbd=a4b, dcd=c-1 >

Subgroups: 1602 in 359 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C4xS3, D12, D12, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, 2+ 1+4, C3xDic3, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, C2xD12, C4oD12, C4oD12, S3xD4, Q8:3S3, C3xC4oD4, C6.D6, C3:D12, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C12:S3, C6xC12, C2xS32, C22xC3:S3, D4oD12, D6.6D6, S3xD12, Dic3:D6, C3xC4oD12, C2xC12:S3, D12:27D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S32, S3xC23, C2xS32, D4oD12, C22xS32, D12:27D6

Permutation representations of D12:27D6
On 24 points - transitive group 24T611
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 15)(16 24)(17 23)(18 22)(19 21)

G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,4)(6,12)(7,11)(8,10)(13,15)(16,24)(17,23)(18,22)(19,21)>;

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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,5)(2,4)(6,12)(7,11)(8,10)(13,15)(16,24)(17,23)(18,22)(19,21) );

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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,15),(16,24),(17,23),(18,22),(19,21)]])

G:=TransitiveGroup(24,611);

45 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C4D4E4F6A6B6C···6G6H6I6J6K12A12B12C12D12E···12J12K12L12M12N
order12222222222333444444666···666661212121212···1212121212
size112666618181818224226666224···41212121222224···412121212

45 irreducible representations

dim111111222222444444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D6D6D6D6D62+ 1+4S32C2xS32C2xS32D4oD12D12:27D6
kernelD12:27D6D6.6D6S3xD12Dic3:D6C3xC4oD12C2xC12:S3C4oD12Dic6C4xS3D12C3:D4C2xC12C32C2xC4C4C22C3C1
# reps144421224242112144

Matrix representation of D12:27D6 in GL4(F13) generated by

7300
101000
98103
89107
,
491211
491112
98103
89103
,
01200
11200
03012
9711
,
101000
7300
36710
5436
G:=sub<GL(4,GF(13))| [7,10,9,8,3,10,8,9,0,0,10,10,0,0,3,7],[4,4,9,8,9,9,8,9,12,11,10,10,11,12,3,3],[0,1,0,9,12,12,3,7,0,0,0,1,0,0,12,1],[10,7,3,5,10,3,6,4,0,0,7,3,0,0,10,6] >;

D12:27D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{27}D_6
% in TeX

G:=Group("D12:27D6");
// GroupNames label

G:=SmallGroup(288,956);
// by ID

G=gap.SmallGroup(288,956);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,100,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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