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## G = C62⋊5D6order 432 = 24·33

### 5th semidirect product of C62 and D6 acting faithfully

Aliases: C625D6, C3⋊S3⋊S4, C3⋊S4⋊S3, (C3×A4)⋊D6, C32⋊(C2×S4), C3.2(S3×S4), C62⋊S3⋊C2, C62⋊C6⋊C2, C32⋊S4⋊C2, C32⋊A4⋊C22, C22⋊(C32⋊D6), (C2×C6).2S32, (C22×C3⋊S3)⋊4S3, SmallGroup(432,523)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C32⋊A4 — C62⋊5D6
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — C62⋊C6 — C62⋊5D6
 Lower central C32⋊A4 — C62⋊5D6
 Upper central C1

Generators and relations for C625D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a2b-1, dad=a-1b-1, cbc-1=a3b-1, bd=db, dcd=c-1 >

Subgroups: 1289 in 134 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, S4, C2×A4, C22×S3, He3, C3×Dic3, S32, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×S4, C32⋊C6, He3⋊C2, C6.D6, C3⋊D12, C3×C3⋊D4, C3×S4, C3⋊S4, S3×A4, C2×S32, C22×C3⋊S3, C32⋊D6, C32⋊A4, Dic3⋊D6, S3×S4, C62⋊S3, C32⋊S4, C62⋊C6, C625D6
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C32⋊D6, S3×S4, C625D6

Character table of C625D6

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 12A 12B size 1 3 9 18 18 27 2 6 24 48 18 18 6 6 12 36 36 72 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 -1 1 1 1 1 1 -1 1 1 1 -1 1 -1 1 -1 linear of order 2 ρ5 2 2 -2 0 0 -2 2 2 -1 -1 0 0 2 2 2 0 0 1 0 0 orthogonal lifted from D6 ρ6 2 2 2 0 0 2 2 2 -1 -1 0 0 2 2 2 0 0 -1 0 0 orthogonal lifted from S3 ρ7 2 2 0 0 -2 0 2 -1 2 -1 -2 0 2 -1 -1 0 1 0 1 0 orthogonal lifted from D6 ρ8 2 2 0 0 2 0 2 -1 2 -1 2 0 2 -1 -1 0 -1 0 -1 0 orthogonal lifted from S3 ρ9 3 -1 3 -1 -1 -1 3 3 0 0 1 1 -1 -1 -1 -1 -1 0 1 1 orthogonal lifted from S4 ρ10 3 -1 -3 1 -1 1 3 3 0 0 1 -1 -1 -1 -1 1 -1 0 1 -1 orthogonal lifted from C2×S4 ρ11 3 -1 3 1 1 -1 3 3 0 0 -1 -1 -1 -1 -1 1 1 0 -1 -1 orthogonal lifted from S4 ρ12 3 -1 -3 -1 1 1 3 3 0 0 -1 1 -1 -1 -1 -1 1 0 -1 1 orthogonal lifted from C2×S4 ρ13 4 4 0 0 0 0 4 -2 -2 1 0 0 4 -2 -2 0 0 0 0 0 orthogonal lifted from S32 ρ14 6 -2 0 2 0 0 -3 0 0 0 0 -2 1 4 -2 -1 0 0 0 1 orthogonal faithful ρ15 6 -2 0 0 2 0 6 -3 0 0 -2 0 -2 1 1 0 -1 0 1 0 orthogonal lifted from S3×S4 ρ16 6 -2 0 0 -2 0 6 -3 0 0 2 0 -2 1 1 0 1 0 -1 0 orthogonal lifted from S3×S4 ρ17 6 6 0 2 0 0 -3 0 0 0 0 2 -3 0 0 -1 0 0 0 -1 orthogonal lifted from C32⋊D6 ρ18 6 -2 0 -2 0 0 -3 0 0 0 0 2 1 4 -2 1 0 0 0 -1 orthogonal faithful ρ19 6 6 0 -2 0 0 -3 0 0 0 0 -2 -3 0 0 1 0 0 0 1 orthogonal lifted from C32⋊D6 ρ20 12 -4 0 0 0 0 -6 0 0 0 0 0 2 -4 2 0 0 0 0 0 orthogonal faithful

Permutation representations of C625D6
On 18 points - transitive group 18T152
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 11 8 12 9 10)(13 17 15)(14 18 16)
(1 16 10)(2 13 8)(3 18 12 5 14 11)(4 15 7 6 17 9)
(1 8)(2 10)(3 7)(4 12)(5 9)(6 11)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(18,152);

On 18 points - transitive group 18T153
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 13 11 2 16 8)(3 15 10 6 14 9)(4 18 7 5 17 12)
(1 11)(2 8)(3 10)(4 7)(5 12)(6 9)

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

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

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

G:=TransitiveGroup(18,153);

On 18 points - transitive group 18T154
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 17 9)(2 14 12)(3 13 8 5 15 7)(4 16 11 6 18 10)
(1 9)(2 12)(3 8)(4 11)(5 7)(6 10)

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

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

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

G:=TransitiveGroup(18,154);

On 18 points - transitive group 18T155
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 10 8 11 9 12)(13 17 15)(14 18 16)
(1 15 11 2 18 7)(3 17 10 6 16 8)(4 14 9 5 13 12)
(1 7)(2 11)(3 9)(4 10)(5 8)(6 12)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(18,155);

Matrix representation of C625D6 in GL6(ℤ)

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

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0],[0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;

C625D6 in GAP, Magma, Sage, TeX

C_6^2\rtimes_5D_6
% in TeX

G:=Group("C6^2:5D6");
// GroupNames label

G:=SmallGroup(432,523);
// by ID

G=gap.SmallGroup(432,523);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,675,353,2524,1271,4548,2287,2659,3989]);
// Polycyclic

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

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