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G = C2×Q83S3order 96 = 25·3

Direct product of C2 and Q83S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q83S3, Q86D6, C6.9C24, D129C22, D6.4C23, C12.23C23, Dic3.9C23, (C2×Q8)⋊8S3, (C6×Q8)⋊6C2, Dic3(C2×Q8), Q8(C2×Dic3), C63(C4○D4), (C2×C4).62D6, (C2×D12)⋊12C2, (C4×S3)⋊5C22, (C3×Q8)⋊6C22, C4.23(C22×S3), C2.10(S3×C23), (C2×C6).67C23, (C2×C12).47C22, C22.32(C22×S3), (C22×S3).30C22, (C2×Dic3).51C22, (S3×C2×C4)⋊5C2, C33(C2×C4○D4), (C2×Q8)(C2×Dic3), SmallGroup(96,213)

Series: Derived Chief Lower central Upper central

C1C6 — C2×Q83S3
C1C3C6D6C22×S3S3×C2×C4 — C2×Q83S3
C3C6 — C2×Q83S3
C1C22C2×Q8

Generators and relations for C2×Q83S3
 G = < a,b,c,d,e | a2=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 338 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×6], C4 [×2], C22, C22 [×12], S3 [×6], C6, C6 [×2], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], Dic3 [×2], C12 [×6], D6 [×6], D6 [×6], C2×C6, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×S3 [×12], D12 [×12], C2×Dic3, C2×C12 [×3], C3×Q8 [×4], C22×S3 [×3], C2×C4○D4, S3×C2×C4 [×3], C2×D12 [×3], Q83S3 [×8], C6×Q8, C2×Q83S3
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, Q83S3 [×2], S3×C23, C2×Q83S3

Character table of C2×Q83S3

 class 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D12E12F
 size 111166666622222223333222444444
ρ1111111111111111111111111111111    trivial
ρ21111-11-1-11-11-1-11-1-111111111-1-11-1-11    linear of order 2
ρ311-1-1-1-1111-11111-1-1-1-111-1-11-11-111-1-1    linear of order 2
ρ411-1-11-1-1-1111-1-1111-1-111-1-11-1-111-11-1    linear of order 2
ρ51111-11-111111-1-11-1-1-1-1-1-1111-1-1-111-1    linear of order 2
ρ61111111-11-11-11-1-11-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ711-1-11-1-111-111-1-1-1111-1-11-11-1-11-11-11    linear of order 2
ρ811-1-1-1-11-1111-11-11-111-1-11-11-11-1-1-111    linear of order 2
ρ911-1-1-1111-1-11-1-1111-11-1-11-11-1-111-11-1    linear of order 2
ρ1011-1-111-1-1-111111-1-1-11-1-11-11-11-111-1-1    linear of order 2
ρ1111111-111-111-1-11-1-11-1-1-1-1111-1-11-1-11    linear of order 2
ρ121111-1-1-1-1-1-11111111-1-1-1-1111111111    linear of order 2
ρ1311-1-111-11-1-11-11-11-11-111-1-11-11-1-1-111    linear of order 2
ρ1411-1-1-111-1-1111-1-1-111-111-1-11-1-11-11-11    linear of order 2
ρ151111-1-1-11-111-11-1-11-1111111111-1-1-1-1    linear of order 2
ρ1611111-11-1-1-111-1-11-1-11111111-1-1-111-1    linear of order 2
ρ1722-2-2000000-1222-2-2-200001-11-11-1-111    orthogonal lifted from D6
ρ182222000000-1-2-22-2-220000-1-1-111-111-1    orthogonal lifted from D6
ρ1922-2-2000000-1-2-2222-200001-111-1-11-11    orthogonal lifted from D6
ρ202222000000-12222220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2122-2-2000000-12-2-2-22200001-111-11-11-1    orthogonal lifted from D6
ρ222222000000-1-22-2-22-20000-1-1-1-1-11111    orthogonal lifted from D6
ρ2322-2-2000000-1-22-22-2200001-11-1111-1-1    orthogonal lifted from D6
ρ242222000000-12-2-22-2-20000-1-1-1111-1-11    orthogonal lifted from D6
ρ252-22-20000002000000-2i2i-2i2i2-2-2000000    complex lifted from C4○D4
ρ262-2-220000002000000-2i-2i2i2i-2-22000000    complex lifted from C4○D4
ρ272-22-200000020000002i-2i2i-2i2-2-2000000    complex lifted from C4○D4
ρ282-2-2200000020000002i2i-2i-2i-2-22000000    complex lifted from C4○D4
ρ294-44-4000000-20000000000-222000000    orthogonal lifted from Q83S3, Schur index 2
ρ304-4-44000000-2000000000022-2000000    orthogonal lifted from Q83S3, Schur index 2

Smallest permutation representation of C2×Q83S3
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

G:=sub<Sym(48)| (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48)>;

G:=Group( (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,24,3,22)(2,23,4,21)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,30,15,32)(14,29,16,31)(17,26,19,28)(18,25,20,27)(33,45,35,47)(34,48,36,46), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,34)(2,33)(3,36)(4,35)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,47)(22,46)(23,45)(24,48) );

G=PermutationGroup([(1,34),(2,35),(3,36),(4,33),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,24,3,22),(2,23,4,21),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,30,15,32),(14,29,16,31),(17,26,19,28),(18,25,20,27),(33,45,35,47),(34,48,36,46)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,45,10),(6,46,11),(7,47,12),(8,48,9),(21,30,25),(22,31,26),(23,32,27),(24,29,28),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,34),(2,33),(3,36),(4,35),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,47),(22,46),(23,45),(24,48)])

C2×Q83S3 is a maximal subgroup of
M4(2).21D6  Q87(C4×S3)  C4⋊C4.150D6  Q8.11D12  Q84D12  D127D4  D12.17D4  C42.126D6  Q86D12  Q87D12  C4⋊C426D6  C6.172- 1+4  D1221D4  D1222D4  C42.233D6  C4220D6  D1210D4  C42.171D6  C42.240D6  D1212D4  D24⋊C22  C6.452- 1+4  C6.1482+ 1+4  C2×S3×C4○D4  D12.39C23
C2×Q83S3 is a maximal quotient of
C6.112+ 1+4  Q87Dic6  Q87D12  C42.131D6  C42.135D6  C42.136D6  C4⋊C4.187D6  C4⋊C426D6  D1221D4  C6.532+ 1+4  C6.772- 1+4  C6.562+ 1+4  C6.782- 1+4  C42.237D6  C42.152D6  C42.153D6  C42.155D6  C42.156D6  C42.240D6  D1212D4  C42.241D6  D129Q8  C42.177D6  C42.178D6  C42.179D6  C2×Q8×Dic3  C6.452- 1+4

Matrix representation of C2×Q83S3 in GL4(𝔽13) generated by

12000
01200
0010
0001
,
12000
01200
00123
0081
,
12000
01200
0050
00128
,
12100
12000
0010
0001
,
01200
12000
0010
00512
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,8,0,0,3,1],[12,0,0,0,0,12,0,0,0,0,5,12,0,0,0,8],[12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,1,5,0,0,0,12] >;

C2×Q83S3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C2xQ8:3S3");
// GroupNames label

G:=SmallGroup(96,213);
// by ID

G=gap.SmallGroup(96,213);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of C2×Q83S3 in TeX

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