non-abelian, soluble, monomial
Aliases: C4.15S3wrC2, C32:Q16:1C2, (C3xC12).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, (C3xC6).7(C2xD4), C2.10(C2xS3wrC2), (C2xC3:S3).31D4, (C4xC3: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, C2xC4, D4, Q8, C32, Dic3, C12, D6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3:S3, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xQ8, C8.C22, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C4oD12, S3xQ8, C32:2C8, C6.D6, D6:S3, C3:D12, C32:2Q8, C32:2Q8, C3xDic6, S3xC12, C4xC3:S3, C32:2SD16, C32:Q16, C32:M4(2), Dic3.D6, D6.D6, C32:Q16:C2
Quotients: C1, C2, C22, D4, C23, C2xD4, C8.C22, S3wrC2, C2xS3wrC2, 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 C2xS3wrC2 |
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 S3wrC2 |
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 S3wrC2 |
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 S3wrC2 |
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 C2xS3wrC2 |
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 C2xS3wrC2 |
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 S3wrC2 |
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 C2xS3wrC2 |
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 |
(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
Export