non-abelian, soluble, monomial, rational
Aliases: C4.5S3≀C2, C32⋊C4⋊1Q8, C32⋊1(C4⋊Q8), (C3×C12).10D4, C3⋊Dic3.28D4, Dic3.D6.5C2, C6.D6.3C22, C2.6(C2×S3≀C2), (C3×C6).3(C2×D4), C3⋊S3.2(C2×Q8), (C4×C32⋊C4).1C2, C3⋊S3.Q8.1C2, (C2×C3⋊S3).3C23, (C4×C3⋊S3).29C22, (C2×C32⋊C4).13C22, SmallGroup(288,870)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊C4⋊Q8
G = < a,b,c,d,e | a3=b3=c4=d4=1, e2=d2, cbc-1=ab=ba, cac-1=a-1b, ad=da, ae=ea, bd=db, ebe-1=a-1b-1, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 504 in 106 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, C32, Dic3, C12, D6, C42, C4⋊C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C4⋊Q8, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×Q8, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C2×C32⋊C4, C3⋊S3.Q8, C4×C32⋊C4, Dic3.D6, C32⋊C4⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4⋊Q8, S3≀C2, C2×S3≀C2, C32⋊C4⋊Q8
Character table of C32⋊C4⋊Q8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 2 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 4 | 4 | 8 | 8 | 24 | 24 | 24 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -1 | 2 | 1 | 0 | -1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -2 | -1 | 0 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | -1 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 2 | -1 | 0 | 1 | 0 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -1 | 2 | -1 | 0 | 1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -2 | 1 | 0 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ21 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 2 | -1 | 0 | -1 | 0 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ22 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | 1 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ23 | 8 | -8 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 18 20)(2 17 19)(3 37 39)(4 40 38)(5 22 24)(6 23 21)(7 41 43)(8 42 44)(9 48 46)(10 47 45)(11 30 32)(12 29 31)(13 36 34)(14 33 35)(15 26 28)(16 25 27)
(2 19 17)(4 38 40)(5 24 22)(7 43 41)(10 45 47)(12 31 29)(13 34 36)(16 27 25)
(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 6 14 15)(2 5 13 16)(3 9 11 8)(4 10 12 7)(17 22 36 25)(18 23 33 26)(19 24 34 27)(20 21 35 28)(29 41 40 47)(30 42 37 48)(31 43 38 45)(32 44 39 46)
(1 7 14 10)(2 8 13 9)(3 5 11 16)(4 6 12 15)(17 42 36 48)(18 41 33 47)(19 44 34 46)(20 43 35 45)(21 31 28 38)(22 30 25 37)(23 29 26 40)(24 32 27 39)
G:=sub<Sym(48)| (1,18,20)(2,17,19)(3,37,39)(4,40,38)(5,22,24)(6,23,21)(7,41,43)(8,42,44)(9,48,46)(10,47,45)(11,30,32)(12,29,31)(13,36,34)(14,33,35)(15,26,28)(16,25,27), (2,19,17)(4,38,40)(5,24,22)(7,43,41)(10,45,47)(12,31,29)(13,34,36)(16,27,25), (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), (1,6,14,15)(2,5,13,16)(3,9,11,8)(4,10,12,7)(17,22,36,25)(18,23,33,26)(19,24,34,27)(20,21,35,28)(29,41,40,47)(30,42,37,48)(31,43,38,45)(32,44,39,46), (1,7,14,10)(2,8,13,9)(3,5,11,16)(4,6,12,15)(17,42,36,48)(18,41,33,47)(19,44,34,46)(20,43,35,45)(21,31,28,38)(22,30,25,37)(23,29,26,40)(24,32,27,39)>;
G:=Group( (1,18,20)(2,17,19)(3,37,39)(4,40,38)(5,22,24)(6,23,21)(7,41,43)(8,42,44)(9,48,46)(10,47,45)(11,30,32)(12,29,31)(13,36,34)(14,33,35)(15,26,28)(16,25,27), (2,19,17)(4,38,40)(5,24,22)(7,43,41)(10,45,47)(12,31,29)(13,34,36)(16,27,25), (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), (1,6,14,15)(2,5,13,16)(3,9,11,8)(4,10,12,7)(17,22,36,25)(18,23,33,26)(19,24,34,27)(20,21,35,28)(29,41,40,47)(30,42,37,48)(31,43,38,45)(32,44,39,46), (1,7,14,10)(2,8,13,9)(3,5,11,16)(4,6,12,15)(17,42,36,48)(18,41,33,47)(19,44,34,46)(20,43,35,45)(21,31,28,38)(22,30,25,37)(23,29,26,40)(24,32,27,39) );
G=PermutationGroup([[(1,18,20),(2,17,19),(3,37,39),(4,40,38),(5,22,24),(6,23,21),(7,41,43),(8,42,44),(9,48,46),(10,47,45),(11,30,32),(12,29,31),(13,36,34),(14,33,35),(15,26,28),(16,25,27)], [(2,19,17),(4,38,40),(5,24,22),(7,43,41),(10,45,47),(12,31,29),(13,34,36),(16,27,25)], [(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)], [(1,6,14,15),(2,5,13,16),(3,9,11,8),(4,10,12,7),(17,22,36,25),(18,23,33,26),(19,24,34,27),(20,21,35,28),(29,41,40,47),(30,42,37,48),(31,43,38,45),(32,44,39,46)], [(1,7,14,10),(2,8,13,9),(3,5,11,16),(4,6,12,15),(17,42,36,48),(18,41,33,47),(19,44,34,46),(20,43,35,45),(21,31,28,38),(22,30,25,37),(23,29,26,40),(24,32,27,39)]])
Matrix representation of C32⋊C4⋊Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,12,0,0,0,0,0,12,0,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C32⋊C4⋊Q8 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_4\rtimes Q_8
% in TeX
G:=Group("C3^2:C4:Q8"); // GroupNames label
G:=SmallGroup(288,870);
// by ID
G=gap.SmallGroup(288,870);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,422,219,100,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^4=1,e^2=d^2,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,a*e=e*a,b*d=d*b,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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