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## G = D12⋊4Dic3order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12⋊4Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4○D12 — D12⋊4Dic3
 Lower central C32 — C3×C6 — C3×C12 — D12⋊4Dic3
 Upper central C1 — C4 — C2×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 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C ··· 6G 6H 6I 8A 8B 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A 24B 24C 24D order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 12 2 2 4 1 1 2 12 18 18 18 18 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - - + + + - - + image C1 C2 C2 C2 C4 C4 S3 S3 D4 D4 Dic3 Dic3 D6 C4×S3 C3⋊D4 D12 C3⋊D4 C4≀C2 S32 S3×Dic3 D6⋊S3 C3⋊D12 D12⋊C4 Q8⋊3Dic3 D12⋊4Dic3 kernel D12⋊4Dic3 C3×C4.Dic3 C4×C3⋊Dic3 C3×C4○D12 C3×Dic6 C3×D12 C4.Dic3 C4○D12 C3×C12 C62 Dic6 D12 C2×C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 4 2 2 4 1 1 1 1 2 2 4

Matrix representation of D124Dic3 in GL4(𝔽5) generated by

 0 0 4 0 0 0 0 1 4 0 2 0 0 1 0 3
,
 0 3 0 0 2 0 0 0 0 4 0 2 1 0 3 0
,
 1 0 3 0 0 0 0 3 3 0 0 0 0 3 0 4
,
 2 0 1 0 0 1 0 3 0 0 3 0 0 0 0 4
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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