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G = D1227D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D1227D6, Dic626D6, C3222+ 1+4, C62.140C23, (C4×S3)⋊2D6, (C2×C12)⋊6D6, C3⋊D47D6, C4○D129S3, (S3×D12)⋊8C2, Dic3⋊D63C2, C32(D4○D12), (C6×C12)⋊8C22, (S3×C12)⋊3C22, D6.6D68C2, (S3×C6).7C23, C6.15(S3×C23), (C3×C6).15C24, D6.8(C22×S3), (C3×D12)⋊32C22, C3⋊D122C22, C6.D62C22, C12⋊S324C22, C12.133(C22×S3), (C3×C12).120C23, (C3×Dic6)⋊31C22, Dic3.7(C22×S3), (C3×Dic3).10C23, (C2×C4)⋊4S32, C4.64(C2×S32), (C2×S32)⋊3C22, C2.17(C22×S32), C22.13(C2×S32), (C3×C4○D12)⋊14C2, (C2×C12⋊S3)⋊19C2, (C3×C3⋊D4)⋊9C22, (C2×C3⋊S3).21C23, (C22×C3⋊S3)⋊6C22, (C2×C6).15(C22×S3), SmallGroup(288,956)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D1227D6
C1C3C32C3×C6S3×C6C2×S32S3×D12 — D1227D6
C32C3×C6 — D1227D6
C1C2C2×C4

Generators and relations for D1227D6
 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 [×9], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×14], S3 [×16], C6 [×2], C6 [×9], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×4], D6 [×28], C2×C6 [×2], C2×C6 [×5], C2×D4 [×9], C4○D4 [×6], C3×S3 [×4], C3⋊S3 [×4], C3×C6, C3×C6, Dic6 [×2], C4×S3 [×4], C4×S3 [×8], D12 [×2], D12 [×20], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×14], 2+ 1+4, C3×Dic3 [×4], C3×C12 [×2], S32 [×4], S3×C6 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×2], C62, C2×D12 [×7], C4○D12 [×2], C4○D12 [×4], S3×D4 [×12], Q83S3 [×4], C3×C4○D4 [×2], C6.D6 [×4], C3⋊D12 [×8], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C3×C3⋊D4 [×4], C12⋊S3 [×4], C6×C12, C2×S32 [×4], C22×C3⋊S3 [×2], D4○D12 [×2], D6.6D6 [×4], S3×D12 [×4], Dic3⋊D6 [×4], C3×C4○D12 [×2], C2×C12⋊S3, D1227D6
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], D4○D12 [×2], C22×S32, D1227D6

Permutation representations of D1227D6
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+4S32C2×S32C2×S32D4○D12D1227D6
kernelD1227D6D6.6D6S3×D12Dic3⋊D6C3×C4○D12C2×C12⋊S3C4○D12Dic6C4×S3D12C3⋊D4C2×C12C32C2×C4C4C22C3C1
# reps144421224242112144

Matrix representation of D1227D6 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] >;

D1227D6 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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