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G = C232D6order 96 = 25·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65D4, C232D6, (C2×C4)⋊2D6, (C2×C6)⋊2D4, C32C22≀C2, (C2×D4)⋊3S3, (C6×D4)⋊8C2, D6⋊C414C2, C2.25(S3×D4), C6.49(C2×D4), (S3×C23)⋊2C2, (C2×C12)⋊7C22, C223(C3⋊D4), (C2×C6).52C23, (C22×C6)⋊3C22, C6.D410C2, (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

C1C2×C6 — C232D6
C1C3C6C2×C6C22×S3S3×C23 — C232D6
C3C2×C6 — C232D6
C1C22C2×D4

Generators and relations for C232D6
 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 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×4], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], Dic3 [×2], C12, D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C22≀C2, D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, C232D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, C232D6

Character table of C232D6

 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 C232D6
On 24 points - transitive group 24T145
Generators in S24
(1 15)(2 13)(3 17)(4 20)(5 24)(6 22)(7 16)(8 14)(9 18)(10 21)(11 19)(12 23)
(1 5)(2 6)(3 4)(7 11)(8 12)(9 10)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 9)(2 7)(3 8)(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 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,145);

C232D6 is a maximal subgroup of
C3⋊C2≀C4  C23.3D12  C23⋊D12  2+ 1+47S3  C4214D6  D1223D4  C4218D6  C4219D6  S3×C22≀C2  C247D6  C248D6  C24.44D6  C24.45D6  C249D6  C6.372+ 1+4  C4⋊C421D6  C6.382+ 1+4  D1219D4  C6.402+ 1+4  D1220D4  C6.422+ 1+4  C6.462+ 1+4  C6.482+ 1+4  C6.1202+ 1+4  C4⋊C428D6  C6.612+ 1+4  C6.1222+ 1+4  C6.622+ 1+4  C6.682+ 1+4  C4222D6  C4223D6  C4224D6  C4228D6  D1211D4  C4230D6  D4×C3⋊D4  C2412D6  (C2×D4)⋊43D6  C6.1452+ 1+4  C6.1462+ 1+4  C232D18  D64D12  C624D4  C628D4  C62.125C23  C6213D4  D6⋊S4  D64D20  C15⋊C22≀C2  D3018D4  D308D4  D3017D4
C232D6 is a maximal quotient of
C24.57D6  C232Dic6  C24.59D6  C24.23D6  C24.25D6  C233D12  (C2×Dic3)⋊Q8  D6⋊C46C4  (C2×C4)⋊3D12  C246D6  D1216D4  D1217D4  Dic617D4  D12.36D4  D12.37D4  Dic6.37D4  C22⋊C4⋊D6  C427D6  D12.14D4  C428D6  D12.15D4  D12⋊D4  Dic6⋊D4  D66SD16  D68SD16  D127D4  Dic6.16D4  D65Q16  D12.17D4  D1218D4  D12.38D4  D12.39D4  D12.40D4  C24.29D6  C24.32D6  C232D18  D64D12  C624D4  C628D4  C62.125C23  C6213D4  D64D20  C15⋊C22≀C2  D3018D4  D308D4  D3017D4

Matrix representation of C232D6 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] >;

C232D6 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 C232D6 in TeX

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