non-abelian, soluble, monomial
Aliases: C4.15S3≀C2, C32⋊Q16⋊1C2, (C3×C12).13D4, Dic3.D6⋊8C2, D6.D6.4C2, C32⋊2SD16⋊3C2, C32⋊1(C8.C22), C3⋊Dic3.4C23, D6⋊S3.6C22, C32⋊M4(2)⋊5C2, C32⋊2C8.1C22, C32⋊2Q8.1C22, (C3×C6).7(C2×D4), C2.10(C2×S3≀C2), (C2×C3⋊S3).31D4, (C4×C3⋊S3).32C22, SmallGroup(288,874)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊Q16⋊C2
G = < a,b,c,d,e | a3=b3=c8=e2=1, d2=c4, ab=ba, cac-1=dad-1=b, eae=cbc-1=a-1, dbd-1=a, ebe=b-1, dcd-1=c-1, ece=c5, de=ed >
Subgroups: 496 in 99 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, S3×Q8, C32⋊2C8, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C32⋊2Q8, C3×Dic6, S3×C12, C4×C3⋊S3, C32⋊2SD16, C32⋊Q16, C32⋊M4(2), Dic3.D6, D6.D6, C32⋊Q16⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C8.C22, S3≀C2, C2×S3≀C2, C32⋊Q16⋊C2
Character table of C32⋊Q16⋊C2
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | |
| size | 1 | 1 | 12 | 18 | 4 | 4 | 2 | 12 | 12 | 12 | 18 | 4 | 4 | 12 | 12 | 36 | 36 | 4 | 4 | 8 | 12 | 12 | 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 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 2 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ12 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | -2 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ13 | 4 | 4 | -2 | 0 | -2 | 1 | 4 | 0 | 0 | -2 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | 4 | 2 | 0 | -2 | 1 | 4 | 0 | 0 | 2 | 0 | -2 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | -2 | -1 | -1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ15 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | -2 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 1 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ16 | 4 | 4 | -2 | 0 | -2 | 1 | -4 | 0 | 0 | 2 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 2 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | 4 | 2 | 0 | -2 | 1 | -4 | 0 | 0 | -2 | 0 | -2 | 1 | -1 | -1 | 0 | 0 | -1 | -1 | 2 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | √-3 | -√-3 | 0 | 0 | -3i | 3i | 0 | √3 | -√3 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | √-3 | -√-3 | 0 | 0 | 3i | -3i | 0 | -√3 | √3 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -√-3 | √-3 | 0 | 0 | -3i | 3i | 0 | -√3 | √3 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -√-3 | √-3 | 0 | 0 | 3i | -3i | 0 | √3 | -√3 | 0 | 0 | complex faithful |
| ρ24 | 8 | -8 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(2 14 46)(4 48 16)(6 10 42)(8 44 12)(17 32 33)(19 35 26)(21 28 37)(23 39 30)
(1 13 45)(3 47 15)(5 9 41)(7 43 11)(18 34 25)(20 27 36)(22 38 29)(24 31 40)
(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 32 13 28)(10 31 14 27)(11 30 15 26)(12 29 16 25)(33 45 37 41)(34 44 38 48)(35 43 39 47)(36 42 40 46)
(2 6)(4 8)(9 41)(10 46)(11 43)(12 48)(13 45)(14 42)(15 47)(16 44)(18 22)(20 24)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)
G:=sub<Sym(48)| (2,14,46)(4,48,16)(6,10,42)(8,44,12)(17,32,33)(19,35,26)(21,28,37)(23,39,30), (1,13,45)(3,47,15)(5,9,41)(7,43,11)(18,34,25)(20,27,36)(22,38,29)(24,31,40), (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), (1,21,5,17)(2,20,6,24)(3,19,7,23)(4,18,8,22)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(33,45,37,41)(34,44,38,48)(35,43,39,47)(36,42,40,46), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(18,22)(20,24)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33)>;
G:=Group( (2,14,46)(4,48,16)(6,10,42)(8,44,12)(17,32,33)(19,35,26)(21,28,37)(23,39,30), (1,13,45)(3,47,15)(5,9,41)(7,43,11)(18,34,25)(20,27,36)(22,38,29)(24,31,40), (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), (1,21,5,17)(2,20,6,24)(3,19,7,23)(4,18,8,22)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(33,45,37,41)(34,44,38,48)(35,43,39,47)(36,42,40,46), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(18,22)(20,24)(25,38)(26,35)(27,40)(28,37)(29,34)(30,39)(31,36)(32,33) );
G=PermutationGroup([[(2,14,46),(4,48,16),(6,10,42),(8,44,12),(17,32,33),(19,35,26),(21,28,37),(23,39,30)], [(1,13,45),(3,47,15),(5,9,41),(7,43,11),(18,34,25),(20,27,36),(22,38,29),(24,31,40)], [(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)], [(1,21,5,17),(2,20,6,24),(3,19,7,23),(4,18,8,22),(9,32,13,28),(10,31,14,27),(11,30,15,26),(12,29,16,25),(33,45,37,41),(34,44,38,48),(35,43,39,47),(36,42,40,46)], [(2,6),(4,8),(9,41),(10,46),(11,43),(12,48),(13,45),(14,42),(15,47),(16,44),(18,22),(20,24),(25,38),(26,35),(27,40),(28,37),(29,34),(30,39),(31,36),(32,33)]])
Matrix representation of C32⋊Q16⋊C2 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 7 | 66 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,0,43,60,0,0,13,30,7,7,0,0,14,66,0,0],[0,0,27,0,0,0,0,27,27,0,0,0,0,27,0,0],[1,72,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
C32⋊Q16⋊C2 in GAP, Magma, Sage, TeX
C_3^2\rtimes Q_{16}\rtimes C_2 % in TeX
G:=Group("C3^2:Q16:C2"); // GroupNames label
G:=SmallGroup(288,874);
// by ID
G=gap.SmallGroup(288,874);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,219,100,675,346,80,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=e^2=1,d^2=c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=b,e*a*e=c*b*c^-1=a^-1,d*b*d^-1=a,e*b*e=b^-1,d*c*d^-1=c^-1,e*c*e=c^5,d*e=e*d>;
// generators/relations
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