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G = D124Dic3order 288 = 25·32

4th semidirect product of D12 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D124Dic3, C62.23D4, Dic64Dic3, C323C4≀C2, (C3×D12)⋊3C4, (C2×C6).8D12, C12.18(C4×S3), (C3×Dic6)⋊3C4, C4○D12.2S3, (C2×C12).79D6, C4.Dic34S3, C4.8(S3×Dic3), C6.38(D6⋊C4), (C3×C12).109D4, C12.9(C2×Dic3), C33(D12⋊C4), C12.95(C3⋊D4), C2.9(D6⋊Dic3), (C6×C12).30C22, C31(Q83Dic3), C4.17(D6⋊S3), C6.8(C6.D4), C22.7(C3⋊D12), (C2×C4).101S32, (C4×C3⋊Dic3)⋊1C2, (C3×C12).28(C2×C4), (C3×C4○D12).1C2, (C2×C6).10(C3⋊D4), (C3×C4.Dic3)⋊15C2, (C3×C6).35(C22⋊C4), SmallGroup(288,216)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D124Dic3
C1C3C32C3×C6C3×C12C6×C12C3×C4○D12 — D124Dic3
C32C3×C6C3×C12 — D124Dic3
C1C4C2×C4

Generators and relations for D124Dic3
 G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 362 in 103 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×6], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×9], C12 [×4], C12 [×3], D6, C2×C6 [×2], C2×C6 [×2], C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3 [×4], C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C4≀C2, C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6, C62, C4.Dic3, C4×Dic3 [×3], C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C2×C3⋊Dic3, C6×C12, D12⋊C4, Q83Dic3, C3×C4.Dic3, C4×C3⋊Dic3, C3×C4○D12, D124Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], C4≀C2, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D12⋊C4, Q83Dic3, D6⋊Dic3, D124Dic3

Permutation representations of D124Dic3
On 24 points - transitive group 24T612
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 15 17 19 21 23)(14 16 18 20 22 24)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 18)

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

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

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

G:=TransitiveGroup(24,612);

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A6B6C···6G6H6I8A8B12A12B12C12D12E···12J12K12L24A24B24C24D
order122233344444444666···666881212121212···12121224242424
size112122241121218181818224···41212121222224···4121212121212

42 irreducible representations

dim1111112222222222224444444
type++++++++--+++--+
imageC1C2C2C2C4C4S3S3D4D4Dic3Dic3D6C4×S3C3⋊D4D12C3⋊D4C4≀C2S32S3×Dic3D6⋊S3C3⋊D12D12⋊C4Q83Dic3D124Dic3
kernelD124Dic3C3×C4.Dic3C4×C3⋊Dic3C3×C4○D12C3×Dic6C3×D12C4.Dic3C4○D12C3×C12C62Dic6D12C2×C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps1111221111112242241111224

Matrix representation of D124Dic3 in GL4(𝔽5) generated by

0040
0001
4020
0103
,
0300
2000
0402
1030
,
1030
0003
3000
0304
,
2010
0103
0030
0004
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,1,4,0,2,0,0,1,0,3],[0,2,0,1,3,0,4,0,0,0,0,3,0,0,2,0],[1,0,3,0,0,0,0,3,3,0,0,0,0,3,0,4],[2,0,0,0,0,1,0,0,1,0,3,0,0,3,0,4] >;

D124Dic3 in GAP, Magma, Sage, TeX

D_{12}\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("D12:4Dic3");
// GroupNames label

G:=SmallGroup(288,216);
// by ID

G=gap.SmallGroup(288,216);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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