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## G = Dic3.S32order 432 = 24·33

### 4th non-split extension by Dic3 of S32 acting via S32/C3×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — Dic3.S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C3×C6.D6 — Dic3.S32
 Lower central C33 — C32×C6 — Dic3.S32
 Upper central C1 — C2

Generators and relations for Dic3.S32
G = < a,b,c,d,e | a6=c2=d3=e2=1, b6=a3, bab-1=cac=eae=a-1, ad=da, cbc=b5, bd=db, be=eb, cd=dc, ece=a3c, ede=d-1 >

Subgroups: 1476 in 218 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22 [×3], S3 [×15], C6, C6 [×2], C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3, Dic3 [×2], Dic3 [×3], C12 [×10], D6 [×13], C2×C6 [×2], C4○D4, C3×S3 [×6], C3⋊S3 [×11], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×2], C4×S3 [×8], D12 [×7], C3⋊D4 [×4], C2×C12 [×2], C3×Q8, C33, C3×Dic3 [×6], C3×Dic3 [×6], C3⋊Dic3, C3×C12 [×3], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×7], C4○D12 [×2], Q83S3, C3×C3⋊S3 [×2], C33⋊C2, C32×C6, C6.D6 [×2], C6.D6 [×3], D6⋊S3, C3⋊D12 [×8], C322Q8, C3×Dic6 [×2], S3×C12 [×4], C4×C3⋊S3, C12⋊S3 [×2], C32×Dic3, C32×Dic3 [×2], C3×C3⋊Dic3, C6×C3⋊S3 [×2], C2×C33⋊C2, D6.D6, D6.6D6 [×2], C3×C6.D6 [×2], C3×C322Q8, C338(C2×C4), C338D4 [×2], C339D4, Dic3.S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×2], Q83S3, C2×S32 [×3], D6.D6, D6.6D6 [×2], S33, Dic3.S32

Permutation representations of Dic3.S32
On 24 points - transitive group 24T1300
Generators in S24
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 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 19)(2 24)(3 17)(4 22)(5 15)(6 20)(7 13)(8 18)(9 23)(10 16)(11 21)(12 14)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)

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

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

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

G:=TransitiveGroup(24,1300);

45 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12A ··· 12H 12I ··· 12P 12Q order 1 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 12 ··· 12 12 ··· 12 12 size 1 1 18 18 54 2 2 2 4 4 4 8 3 3 6 6 18 2 2 2 4 4 4 8 18 18 18 18 6 ··· 6 12 ··· 12 36

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 C4○D12 S32 Q8⋊3S3 C2×S32 D6.D6 D6.6D6 S33 Dic3.S32 kernel Dic3.S32 C3×C6.D6 C3×C32⋊2Q8 C33⋊8(C2×C4) C33⋊8D4 C33⋊9D4 C6.D6 C32⋊2Q8 C3×Dic3 C3⋊Dic3 C2×C3⋊S3 C33 C32 Dic3 C32 C6 C3 C3 C2 C1 # reps 1 2 1 1 2 1 2 1 6 1 2 2 8 3 1 3 2 4 1 1

Matrix representation of Dic3.S32 in GL8(ℤ)

 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 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 0 1 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0
,
 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0
,
 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1
,
 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0] >;

Dic3.S32 in GAP, Magma, Sage, TeX

{\rm Dic}_3.S_3^2
% in TeX

G:=Group("Dic3.S3^2");
// GroupNames label

G:=SmallGroup(432,612);
// by ID

G=gap.SmallGroup(432,612);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,135,58,298,2028,14118]);
// Polycyclic

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

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