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G = C23⋊2D6  order 96 = 25·3

1st semidirect product of C23 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊5D4, C23⋊2D6, (C2×C4)⋊2D6, (C2×C6)⋊2D4, C3⋊2C22≀C2, (C2×D4)⋊3S3, (C6×D4)⋊8C2, D6⋊C4⋊14C2, C2.25(S3×D4), C6.49(C2×D4), (S3×C23)⋊2C2, (C2×C12)⋊7C22, C22⋊3(C3⋊D4), (C2×C6).52C23, (C22×C6)⋊3C22, C6.D4⋊10C2, (C2×Dic3)⋊2C22, C22.59(C22×S3), (C22×S3).25C22, (C2×C3⋊D4)⋊4C2, C2.13(C2×C3⋊D4), SmallGroup(96,144)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C23⋊2D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C23⋊2D6
Lower central
C3 — C2×C6 — C23⋊2D6
Upper central
C1 — C22 — C2×D4

Generators and relations for C23⋊2D6
 G = < a,b,c,d,e | a2=b2=c2=d6=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 354 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C2×D4, C24, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22≀C2, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C23⋊2D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, C23⋊2D6

Character table of C23⋊2D6

 class 12A2B2C2D2E2F2G2H2I2J34A4B4C6A6B6C6D6E6F6G12A12B
 size 11112246666241212222444444
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-111-1111-1111-1-1-1-111    linear of order 2
ρ31111-1-11-111-11-1-11111-1-111-1-1    linear of order 2
ρ4111111-111111-1-1-111111-1-1-1-1    linear of order 2
ρ51111111-1-1-1-111-1-1111111111    linear of order 2
ρ61111-1-1-11-1-1111-11111-1-1-1-111    linear of order 2
ρ71111-1-111-1-111-11-1111-1-111-1-1    linear of order 2
ρ8111111-1-1-1-1-11-11111111-1-1-1-1    linear of order 2
ρ9222222-20000-1-200-1-1-1-1-11111    orthogonal lifted from D6
ρ102-2-22000-20022000-2-22000000    orthogonal lifted from D4
ρ112-2-22000200-22000-2-22000000    orthogonal lifted from D4
ρ122-22-200002-2020002-2-2000000    orthogonal lifted from D4
ρ132222-2-220000-1-200-1-1-111-1-111    orthogonal lifted from D6
ρ142-22-20000-22020002-2-2000000    orthogonal lifted from D4
ρ152222-2-2-20000-1200-1-1-11111-1-1    orthogonal lifted from D6
ρ1622-2-22-2000002000-22-2-220000    orthogonal lifted from D4
ρ1722222220000-1200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1822-2-2-22000002000-22-22-20000    orthogonal lifted from D4
ρ1922-2-2-2200000-10001-11-11√-3-√-3√-3-√-3    complex lifted from C3⋊D4
ρ2022-2-22-200000-10001-111-1-√-3√-3√-3-√-3    complex lifted from C3⋊D4
ρ2122-2-22-200000-10001-111-1√-3-√-3-√-3√-3    complex lifted from C3⋊D4
ρ2222-2-2-2200000-10001-11-11-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ234-44-40000000-2000-222000000    orthogonal lifted from S3×D4
ρ244-4-440000000-200022-2000000    orthogonal lifted from S3×D4

Permutation representations of C23⋊2D6
►On 24 points - transitive group 24T145
Generators in S24
(1 17)(2 15)(3 13)(4 19)(5 23)(6 21)(7 16)(8 14)(9 18)(10 20)(11 24)(12 22)

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

(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 14)(15 18)(16 17)(19 24)(20 23)(21 22)
 
G:=sub<Sym(24)| (1,17)(2,15)(3,13)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,20)(11,24)(12,22), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22)>;
 
G:=Group( (1,17)(2,15)(3,13)(4,19)(5,23)(6,21)(7,16)(8,14)(9,18)(10,20)(11,24)(12,22), (1,4)(2,5)(3,6)(7,11)(8,12)(9,10)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,8)(2,9)(3,7)(4,12)(5,10)(6,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22) );
 
G=PermutationGroup([[(1,17),(2,15),(3,13),(4,19),(5,23),(6,21),(7,16),(8,14),(9,18),(10,20),(11,24),(12,22)], [(1,4),(2,5),(3,6),(7,11),(8,12),(9,10),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,8),(2,9),(3,7),(4,12),(5,10),(6,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14),(15,18),(16,17),(19,24),(20,23),(21,22)]])
 
G:=TransitiveGroup(24,145);
 

C23⋊2D6 is a maximal subgroup of
 C3⋊C2≀C4  C23.3D12  C23⋊D12  2+ 1+4⋊7S3  C42⋊14D6  D12⋊23D4  C42⋊18D6  C42⋊19D6  S3×C22≀C2  C24⋊7D6  C24⋊8D6  C24.44D6  C24.45D6  C24⋊9D6  C6.372+ 1+4  C4⋊C4⋊21D6  C6.382+ 1+4  D12⋊19D4  C6.402+ 1+4  D12⋊20D4  C6.422+ 1+4  C6.462+ 1+4  C6.482+ 1+4  C6.1202+ 1+4  C4⋊C4⋊28D6  C6.612+ 1+4  C6.1222+ 1+4  C6.622+ 1+4  C6.682+ 1+4  C42⋊22D6  C42⋊23D6  C42⋊24D6  C42⋊28D6  D12⋊11D4  C42⋊30D6  D4×C3⋊D4  C24⋊12D6  (C2×D4)⋊43D6  C6.1452+ 1+4  C6.1462+ 1+4  C23⋊2D18  D6⋊4D12  C62⋊4D4  C62⋊8D4  C62.125C23  C62⋊13D4  D6⋊S4  D6⋊4D20  C15⋊C22≀C2  D30⋊18D4  D30⋊8D4  D30⋊17D4
C23⋊2D6 is a maximal quotient of
 C24.57D6  C23⋊2Dic6  C24.59D6  C24.23D6  C24.25D6  C23⋊3D12  (C2×Dic3)⋊Q8  D6⋊C4⋊6C4  (C2×C4)⋊3D12  C24⋊6D6  D12⋊16D4  D12⋊17D4  Dic6⋊17D4  D12.36D4  D12.37D4  Dic6.37D4  C22⋊C4⋊D6  C42⋊7D6  D12.14D4  C42⋊8D6  D12.15D4  D12⋊D4  Dic6⋊D4  D6⋊6SD16  D6⋊8SD16  D12⋊7D4  Dic6.16D4  D6⋊5Q16  D12.17D4  D12⋊18D4  D12.38D4  D12.39D4  D12.40D4  C24.29D6  C24.32D6  C23⋊2D18  D6⋊4D12  C62⋊4D4  C62⋊8D4  C62.125C23  C62⋊13D4  D6⋊4D20  C15⋊C22≀C2  D30⋊18D4  D30⋊8D4  D30⋊17D4

Matrix representation of C23⋊2D6 ►in GL4(𝔽13) generated by

2400
91100
0003
0090
,
12000
01200
0010
0001
,
1000
0100
00120
00012
,
1100
12000
00120
0001
,
1100
01200
0010
00012
G:=sub<GL(4,GF(13))| [2,9,0,0,4,11,0,0,0,0,0,9,0,0,3,0],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,0,12] >;
 

C23⋊2D6 in GAP, Magma, Sage, TeX 

C_2^3\rtimes_2D_6
 
% in TeX
 
G:=Group("C2^3:2D6");
 
// GroupNames label
 
G:=SmallGroup(96,144);
 
// by ID
 
G=gap.SmallGroup(96,144);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C23⋊2D6 in TeX

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