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G = C2×S3×Q8order 96 = 25·3

Direct product of C2, S3 and Q8

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

Aliases: C2×S3×Q8, C6.8C24, C12.22C23, Dic69C22, D6.10C23, Dic3.5C23, C62(C2×Q8), (C6×Q8)⋊5C2, C32(C22×Q8), (C2×C4).61D6, C2.9(S3×C23), (C3×Q8)⋊5C22, (C2×Dic6)⋊13C2, C4.22(C22×S3), (C2×C6).66C23, (C4×S3).13C22, (C2×C12).46C22, C22.31(C22×S3), (C22×S3).36C22, (C2×Dic3).44C22, (S3×C2×C4).6C2, SmallGroup(96,212)

Series: Derived Chief Lower central Upper central

C1C6 — C2×S3×Q8
C1C3C6D6C22×S3S3×C2×C4 — C2×S3×Q8
C3C6 — C2×S3×Q8
C1C22C2×Q8

Generators and relations for C2×S3×Q8
 G = < a,b,c,d,e | a2=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 274 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4 [×3], C2×C4 [×15], Q8 [×4], Q8 [×12], C23, Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic6 [×12], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×3], C3×Q8 [×4], C22×S3, C22×Q8, C2×Dic6 [×3], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, C2×S3×Q8
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, C22×S3 [×7], C22×Q8, S3×Q8 [×2], S3×C23, C2×S3×Q8

Character table of C2×S3×Q8

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A12B12C12D12E12F
 size 111133332222222666666222444444
ρ1111111111111111111111111111111    trivial
ρ21-1-11-111-1111-11-1-111-1-11-1-1-11-1-1111-1    linear of order 2
ρ31111111111-1-1-1-11-1-11-11-1111-111-1-1-1    linear of order 2
ρ41-1-11-111-111-11-11-1-1-1-1111-1-111-11-1-11    linear of order 2
ρ51-1-111-1-11111-11-1-1-1-111-11-1-11-1-1111-1    linear of order 2
ρ61111-1-1-1-11111111-1-1-1-1-1-1111111111    linear of order 2
ρ71-1-111-1-1111-11-11-1111-1-1-1-1-111-11-1-11    linear of order 2
ρ81111-1-1-1-111-1-1-1-1111-11-11111-111-1-1-1    linear of order 2
ρ9111111111-1-1-111-1-11-11-1-1111-1-1-11-11    linear of order 2
ρ101-1-11-111-11-1-111-11-111-1-11-1-1111-11-1-1    linear of order 2
ρ11111111111-111-1-1-11-1-1-1-111111-1-1-11-1    linear of order 2
ρ121-1-11-111-11-11-1-1111-111-1-1-1-11-11-1-111    linear of order 2
ρ131-1-111-1-111-1-111-111-1-111-1-1-1111-11-1-1    linear of order 2
ρ141111-1-1-1-11-1-1-111-11-11-111111-1-1-11-11    linear of order 2
ρ151-1-111-1-111-11-1-111-11-1-111-1-11-11-1-111    linear of order 2
ρ161111-1-1-1-11-111-1-1-1-11111-11111-1-1-11-1    linear of order 2
ρ1722220000-1-222-2-2-2000000-1-1-1-1111-11    orthogonal lifted from D6
ρ182-2-220000-1-22-2-22200000011-11-111-1-1    orthogonal lifted from D6
ρ1922220000-1222222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ202-2-220000-1-2-222-2200000011-1-1-11-111    orthogonal lifted from D6
ρ2122220000-1-2-2-222-2000000-1-1-1111-11-1    orthogonal lifted from D6
ρ2222220000-12-2-2-2-22000000-1-1-11-1-1111    orthogonal lifted from D6
ρ232-2-220000-12-22-22-200000011-1-11-111-1    orthogonal lifted from D6
ρ242-2-220000-122-22-2-200000011-111-1-1-11    orthogonal lifted from D6
ρ2522-2-222-2-22000000000000-22-2000000    symplectic lifted from Q8, Schur index 2
ρ262-22-2-22-2220000000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ272-22-22-22-220000000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ2822-2-2-2-2222000000000000-22-2000000    symplectic lifted from Q8, Schur index 2
ρ294-44-40000-2000000000000-222000000    symplectic lifted from S3×Q8, Schur index 2
ρ3044-4-40000-20000000000002-22000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of C2×S3×Q8
On 48 points
Generators in S48
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 38)(10 39)(11 40)(12 37)(13 25)(14 26)(15 27)(16 28)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 35 38)(30 36 39)(31 33 40)(32 34 37)
(1 35)(2 36)(3 33)(4 34)(5 38)(6 39)(7 40)(8 37)(9 29)(10 30)(11 31)(12 32)(13 23)(14 24)(15 21)(16 22)(17 41)(18 42)(19 43)(20 44)(25 47)(26 48)(27 45)(28 46)
(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 43 3 41)(2 42 4 44)(5 13 7 15)(6 16 8 14)(9 47 11 45)(10 46 12 48)(17 35 19 33)(18 34 20 36)(21 38 23 40)(22 37 24 39)(25 31 27 29)(26 30 28 32)

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

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

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

C2×S3×Q8 is a maximal subgroup of
(S3×Q8)⋊C4  Q83D12  D6⋊Q16  D68SD16  D65Q16  C42.125D6  Q86D12  C6.162- 1+4  Dic621D4  Dic622D4  C42.141D6  Dic610D4  C42.171D6  D128Q8  C6.1072- 1+4
C2×S3×Q8 is a maximal quotient of
C6.102+ 1+4  Dic610Q8  C42.232D6  D1210Q8  (Q8×Dic3)⋊C2  C6.752- 1+4  Dic621D4  C6.512+ 1+4  C6.1182+ 1+4  C6.522+ 1+4  Dic67Q8  C42.236D6  C42.148D6  D127Q8  Dic68Q8  Dic69Q8  D128Q8  C42.241D6  C42.174D6  D129Q8

Matrix representation of C2×S3×Q8 in GL4(𝔽13) generated by

12000
01200
00120
00012
,
121200
1000
0010
0001
,
12000
1100
00120
00012
,
1000
0100
00128
0031
,
1000
0100
0050
00118
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,3,0,0,8,1],[1,0,0,0,0,1,0,0,0,0,5,11,0,0,0,8] >;

C2×S3×Q8 in GAP, Magma, Sage, TeX

C_2\times S_3\times Q_8
% in TeX

G:=Group("C2xS3xQ8");
// GroupNames label

G:=SmallGroup(96,212);
// by ID

G=gap.SmallGroup(96,212);
# by ID

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

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

Export

Character table of C2×S3×Q8 in TeX

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