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G = C6210D6order 432 = 24·33

10th semidirect product of C62 and D6 acting faithfully

non-abelian, soluble, monomial

Aliases: C6210D6, A42S32, C3⋊S42S3, C3⋊S32S4, C32(S3×S4), (C3×A4)⋊6D6, C325(C2×S4), (C32×A4)⋊4C22, C22⋊(C324D6), (C2×C6)⋊2S32, (C3×C3⋊S4)⋊2C2, (A4×C3⋊S3)⋊2C2, (C22×C3⋊S3)⋊7S3, SmallGroup(432,748)

Series: Derived Chief Lower central Upper central

C1C22C32×A4 — C6210D6
C1C22C2×C6C62C32×A4C3×C3⋊S4 — C6210D6
C32×A4 — C6210D6
C1

Generators and relations for C6210D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a2b3, dad=ab3, cbc-1=a3b-1, dbd=b-1, dcd=c-1 >

Subgroups: 1444 in 164 conjugacy classes, 20 normal (10 characteristic)
C1, C2 [×5], C3 [×2], C3 [×5], C4 [×2], C22, C22 [×6], S3 [×12], C6 [×6], C2×C4, D4 [×4], C23 [×2], C32, C32 [×6], Dic3 [×2], C12 [×2], A4, A4 [×3], D6 [×13], C2×C6 [×2], C2×C6 [×3], C2×D4, C3×S3 [×9], C3⋊S3, C3⋊S3 [×3], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], S4 [×4], C2×A4, C22×S3 [×5], C33, C3×Dic3 [×2], S32 [×4], C3×A4 [×2], C3×A4 [×4], S3×C6 [×2], C2×C3⋊S3 [×2], C62, S3×D4 [×2], C2×S4, C3×C3⋊S3 [×3], C6.D6, C3⋊D12 [×2], C3×C3⋊D4 [×2], C3×S4 [×4], C3⋊S4 [×2], S3×A4 [×3], C2×S32, C22×C3⋊S3, C324D6, C32×A4, Dic3⋊D6, S3×S4 [×2], C3×C3⋊S4 [×2], A4×C3⋊S3, C6210D6
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], S4, S32 [×3], C2×S4, C324D6, S3×S4 [×2], C6210D6

Character table of C6210D6

 class 12A2B2C2D2E3A3B3C3D3E3F3G3H4A4B6A6B6C6D6E6F12A12B
 size 139181827224816161616181866123636723636
ρ1111111111111111111111111    trivial
ρ211-1-11-1111111111-1111-11-1-11    linear of order 2
ρ311-11-1-111111111-111111-1-11-1    linear of order 2
ρ4111-1-1111111111-1-1111-1-11-1-1    linear of order 2
ρ5220200-12-12-1-12-102-12-1-100-10    orthogonal lifted from S3
ρ622-200-2222-1-1-1-1-10022200100    orthogonal lifted from D6
ρ7220-200-12-12-1-12-10-2-12-110010    orthogonal lifted from D6
ρ82200-202-1-12-1-1-12-202-1-101001    orthogonal lifted from D6
ρ9222002222-1-1-1-1-10022200-100    orthogonal lifted from S3
ρ102200202-1-12-1-1-12202-1-10-100-1    orthogonal lifted from S3
ρ113-13-1-1-13330000011-1-1-1-1-1011    orthogonal lifted from S4
ρ123-1-3-11133300000-11-1-1-1-1101-1    orthogonal lifted from C2×S4
ρ133-1-31-11333000001-1-1-1-11-10-11    orthogonal lifted from C2×S4
ρ143-1311-133300000-1-1-1-1-1110-1-1    orthogonal lifted from S4
ρ15440000-24-2-211-2100-24-200000    orthogonal lifted from S32
ρ16440000-2-21411-2-200-2-2100000    orthogonal lifted from S32
ρ174400004-2-2-2111-2004-2-200000    orthogonal lifted from S32
ρ18440000-2-21-2-1+3-3/2-1-3-3/21100-2-2100000    complex lifted from C324D6
ρ19440000-2-21-2-1-3-3/2-1+3-3/21100-2-2100000    complex lifted from C324D6
ρ206-200-206-3-30000020-2110100-1    orthogonal lifted from S3×S4
ρ216-20200-36-3000000-21-21-10010    orthogonal lifted from S3×S4
ρ226-20-200-36-300000021-21100-10    orthogonal lifted from S3×S4
ρ236-200206-3-300000-20-2110-1001    orthogonal lifted from S3×S4
ρ2412-40000-6-63000000022-100000    orthogonal faithful

Permutation representations of C6210D6
On 24 points - transitive group 24T1340
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 16 5 14 3 18)(2 17 6 15 4 13)(7 22 9 24 11 20)(8 23 10 19 12 21)
(2 13 18 6 15 16)(3 5)(4 17 14)(7 8 24 11 10 22)(9 12 20)(19 21)
(1 23)(2 7)(3 19)(4 9)(5 21)(6 11)(8 16)(10 18)(12 14)(13 22)(15 24)(17 20)

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

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

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

G:=TransitiveGroup(24,1340);

Matrix representation of C6210D6 in GL7(ℤ)

0100000
-1-100000
0010000
0001000
0000001
0000-1-1-1
0000100
,
1000000
0100000
00-11000
00-10000
0000010
0000100
0000-1-1-1
,
1000000
-1-100000
0001000
0010000
0000-100
0000111
00000-10
,
-1000000
0-100000
000-1000
00-10000
0000100
0000010
0000-1-1-1

G:=sub<GL(7,Integers())| [0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,-1,0,0,0,0,0,0,-1],[1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,-1] >;

C6210D6 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{10}D_6
% in TeX

G:=Group("C6^2:10D6");
// GroupNames label

G:=SmallGroup(432,748);
// by ID

G=gap.SmallGroup(432,748);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,675,346,1271,9077,2287,5298,3989]);
// 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^3,d*a*d=a*b^3,c*b*c^-1=a^3*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Character table of C6210D6 in TeX

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