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G = C625D6order 432 = 24·33

5th semidirect product of C62 and D6 acting faithfully

non-abelian, soluble, monomial, rational

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

C1C2×C6C32⋊A4 — C625D6
C1C22C2×C6C62C32⋊A4C62⋊C6 — C625D6
C32⋊A4 — C625D6
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 [×5], C3, C3 [×3], C4 [×2], C22, C22 [×6], S3 [×10], C6 [×6], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], Dic3 [×2], C12 [×2], A4 [×2], D6 [×13], C2×C6, C2×C6 [×3], C2×D4, C3×S3 [×5], C3⋊S3, C3⋊S3 [×2], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], S4 [×3], C2×A4, C22×S3 [×4], He3, C3×Dic3 [×2], S32 [×3], C3×A4, C3×A4, S3×C6 [×2], C2×C3⋊S3 [×2], C62, S3×D4 [×2], C2×S4, C32⋊C6 [×2], He3⋊C2, C6.D6, C3⋊D12 [×2], C3×C3⋊D4 [×2], C3×S4 [×2], 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 [×3], C22, S3 [×2], D6 [×2], S4, S32, C2×S4, C32⋊D6, S3×S4, C625D6

Character table of C625D6

 class 12A2B2C2D2E3A3B3C3D4A4B6A6B6C6D6E6F12A12B
 size 139181827262448181866123636723636
ρ111111111111111111111    trivial
ρ211-11-1-11111-111111-1-1-11    linear of order 2
ρ3111-1-111111-1-1111-1-11-1-1    linear of order 2
ρ411-1-11-111111-1111-11-11-1    linear of order 2
ρ522-200-222-1-10022200100    orthogonal lifted from D6
ρ622200222-1-10022200-100    orthogonal lifted from S3
ρ72200-202-12-1-202-1-101010    orthogonal lifted from D6
ρ82200202-12-1202-1-10-10-10    orthogonal lifted from S3
ρ93-13-1-1-1330011-1-1-1-1-1011    orthogonal lifted from S4
ρ103-1-31-1133001-1-1-1-11-101-1    orthogonal lifted from C2×S4
ρ113-1311-13300-1-1-1-1-1110-1-1    orthogonal lifted from S4
ρ123-1-3-1113300-11-1-1-1-110-11    orthogonal lifted from C2×S4
ρ134400004-2-21004-2-200000    orthogonal lifted from S32
ρ146-20200-30000-214-2-10001    orthogonal faithful
ρ156-200206-300-20-2110-1010    orthogonal lifted from S3×S4
ρ166-200-206-30020-211010-10    orthogonal lifted from S3×S4
ρ17660200-300002-300-1000-1    orthogonal lifted from C32⋊D6
ρ186-20-200-30000214-21000-1    orthogonal faithful
ρ19660-200-30000-2-30010001    orthogonal lifted from C32⋊D6
ρ2012-40000-6000002-4200000    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 12 8 10 9 11)(13 17 15)(14 18 16)
(1 18 9)(2 15 12)(3 14 8 5 16 7)(4 17 11 6 13 10)
(1 12)(2 9)(3 11)(4 8)(5 10)(6 7)(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)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,18,9)(2,15,12)(3,14,8,5,16,7)(4,17,11,6,13,10), (1,12)(2,9)(3,11)(4,8)(5,10)(6,7)(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)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,18,9)(2,15,12)(3,14,8,5,16,7)(4,17,11,6,13,10), (1,12)(2,9)(3,11)(4,8)(5,10)(6,7)(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),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,18,9),(2,15,12),(3,14,8,5,16,7),(4,17,11,6,13,10)], [(1,12),(2,9),(3,11),(4,8),(5,10),(6,7),(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 11 8 12 9 10)(13 17 15)(14 18 16)
(1 14 11 2 17 9)(3 16 10 6 15 7)(4 13 8 5 18 12)
(1 11)(2 9)(3 10)(4 8)(5 12)(6 7)

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

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

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

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 11 8 12 9 10)(13 17 15)(14 18 16)
(1 13 8)(2 16 10)(3 15 7 5 17 9)(4 18 12 6 14 11)
(1 8)(2 10)(3 7)(4 12)(5 9)(6 11)

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

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

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

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

G:=TransitiveGroup(18,155);

Matrix representation of C625D6 in GL6(ℤ)

-100000
0-10000
001100
00-1000
000001
0000-1-1
,
110000
-100000
00-1-100
001000
000011
0000-10
,
000010
0000-1-1
100000
-1-10000
001000
00-1-100
,
000010
000001
001000
000100
100000
010000

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

Export

Character table of C625D6 in TeX

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