Copied to
clipboard

G = C2×Dic3⋊D6order 288 = 25·32

Direct product of C2 and Dic3⋊D6

direct product, metabelian, supersoluble, monomial, rational

Aliases: C2×Dic3⋊D6, C623C23, C235S32, C63(S3×D4), C3⋊D49D6, (C22×C6)⋊8D6, (S3×C6)⋊5C23, D64(C22×S3), C328(C22×D4), (C2×Dic3)⋊16D6, (C22×S3)⋊13D6, C6.36(S3×C23), (C3×C6).36C24, (C2×C62)⋊7C22, (C6×Dic3)⋊9C22, (C3×Dic3)⋊3C23, Dic33(C22×S3), C3⋊D1218C22, C6.D612C22, C34(C2×S3×D4), C3⋊S34(C2×D4), C224(C2×S32), (C3×C6)⋊7(C2×D4), (C2×C3⋊S3)⋊18D4, (C22×S32)⋊9C2, (C6×C3⋊D4)⋊7C2, (C2×S32)⋊12C22, (C2×C3⋊D4)⋊11S3, (C23×C3⋊S3)⋊5C2, (C2×C3⋊S3)⋊5C23, (S3×C2×C6)⋊12C22, (C2×C6)⋊5(C22×S3), C2.36(C22×S32), (C2×C6.D6)⋊6C2, (C2×C3⋊D12)⋊22C2, (C3×C3⋊D4)⋊13C22, (C22×C3⋊S3)⋊14C22, SmallGroup(288,977)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×Dic3⋊D6
C1C3C32C3×C6S3×C6C2×S32C22×S32 — C2×Dic3⋊D6
C32C3×C6 — C2×Dic3⋊D6
C1C22C23

Generators and relations for C2×Dic3⋊D6
 G = < a,b,c,d,e | a2=b6=d6=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=b3c, ce=ec, ede=d-1 >

Subgroups: 2370 in 539 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×36], S3 [×24], C6 [×6], C6 [×15], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, Dic3 [×4], C12 [×4], D6 [×4], D6 [×84], C2×C6 [×6], C2×C6 [×19], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×4], C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×2], C3⋊D4 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×50], C22×C6 [×2], C22×C6 [×3], C22×D4, C3×Dic3 [×4], S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], S3×C2×C4 [×2], C2×D12 [×2], S3×D4 [×16], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×C23 [×5], C6.D6 [×4], C3⋊D12 [×8], C6×Dic3 [×2], C3×C3⋊D4 [×8], C2×S32 [×4], C2×S32 [×4], S3×C2×C6 [×2], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, C2×S3×D4 [×2], C2×C6.D6, C2×C3⋊D12 [×2], Dic3⋊D6 [×8], C6×C3⋊D4 [×2], C22×S32, C23×C3⋊S3, C2×Dic3⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C22×S3 [×14], C22×D4, S32, S3×D4 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×D4 [×2], Dic3⋊D6 [×2], C22×S32, C2×Dic3⋊D6

Permutation representations of C2×Dic3⋊D6
On 24 points - transitive group 24T671
Generators in S24
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 22 4 19)(2 21 5 24)(3 20 6 23)(7 14 10 17)(8 13 11 16)(9 18 12 15)
(1 7 5 11 3 9)(2 8 6 12 4 10)(13 20 15 22 17 24)(14 21 16 23 18 19)
(1 5)(2 4)(8 12)(9 11)(13 15)(16 18)(19 21)(22 24)

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

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

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

G:=TransitiveGroup(24,671);

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B3C4A4B4C4D6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222222222222233344446···66···6666612121212
size11112266669999181822466662···24···41212121212121212

48 irreducible representations

dim11111112222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D6D6D6D6S32S3×D4C2×S32Dic3⋊D6
kernelC2×Dic3⋊D6C2×C6.D6C2×C3⋊D12Dic3⋊D6C6×C3⋊D4C22×S32C23×C3⋊S3C2×C3⋊D4C2×C3⋊S3C2×Dic3C3⋊D4C22×S3C22×C6C23C6C22C2
# reps11282112428221434

Matrix representation of C2×Dic3⋊D6 in GL6(𝔽13)

1200000
0120000
0012000
0001200
000010
000001
,
1210000
1200000
0012000
0001200
000010
000001
,
010000
100000
001500
00101200
000010
000001
,
100000
010000
0012000
003100
000001
00001212
,
0120000
1200000
0012000
0001200
000001
000010

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,10,0,0,0,0,5,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,3,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×Dic3⋊D6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\rtimes D_6
% in TeX

G:=Group("C2xDic3:D6");
// GroupNames label

G:=SmallGroup(288,977);
// by ID

G=gap.SmallGroup(288,977);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,346,1356,9414]);
// Polycyclic

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

׿
×
𝔽