non-abelian, soluble, monomial
Aliases: C32:D8:5C2, C4.20S3wrC2, C32:Q16:5C2, (C3xC12).20D4, C32:1(C4oD8), D6.D6:1C2, C32:2SD16:7C2, C3:Dic3.1C23, D6:S3.5C22, C32:2C8.9C22, C32:2Q8.6C22, C2.7(C2xS3wrC2), C3:S3:3C8:7C2, (C3xC6).4(C2xD4), (C2xC3:S3).28D4, (C4xC3:S3).60C22, SmallGroup(288,871)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:D8:5C2
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, cac-1=b, dad=eae=cbc-1=a-1, bd=db, ebe=b-1, dcd=c-1, ce=ec, ede=c4d >
Subgroups: 528 in 102 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, Q8, C32, Dic3, C12, D6, C2xC6, C2xC8, D8, SD16, Q16, C4oD4, C3xS3, C3:S3, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C4oD8, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C4oD12, C32:2C8, D6:S3, C3:D12, C32:2Q8, S3xC12, C4xC3:S3, C32:D8, C32:2SD16, C32:Q16, C3:S3:3C8, D6.D6, C32:D8:5C2
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD8, S3wrC2, C2xS3wrC2, C32:D8:5C2
Character table of C32:D8:5C2
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 12 | 12 | 18 | 4 | 4 | 1 | 1 | 12 | 12 | 18 | 4 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | |
| 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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √2 | √-2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4oD8 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | √2 | -√2 | -√-2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4oD8 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | -√2 | √2 | -√-2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4oD8 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √2 | -√2 | √-2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4oD8 |
| ρ15 | 4 | 4 | -2 | 0 | 0 | -2 | 1 | 4 | 4 | -2 | 0 | 0 | -2 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | 0 | 1 | 0 | orthogonal lifted from S3wrC2 |
| ρ16 | 4 | 4 | 2 | 0 | 0 | -2 | 1 | 4 | 4 | 2 | 0 | 0 | -2 | 1 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | -1 | 0 | -1 | 0 | orthogonal lifted from S3wrC2 |
| ρ17 | 4 | 4 | 0 | -2 | 0 | 1 | -2 | -4 | -4 | 0 | 2 | 0 | 1 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | 0 | -1 | 0 | -1 | orthogonal lifted from C2xS3wrC2 |
| ρ18 | 4 | 4 | 0 | 2 | 0 | 1 | -2 | -4 | -4 | 0 | -2 | 0 | 1 | -2 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | 0 | 1 | 0 | 1 | orthogonal lifted from C2xS3wrC2 |
| ρ19 | 4 | 4 | -2 | 0 | 0 | -2 | 1 | -4 | -4 | 2 | 0 | 0 | -2 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | 0 | -1 | 0 | orthogonal lifted from C2xS3wrC2 |
| ρ20 | 4 | 4 | 0 | 2 | 0 | 1 | -2 | 4 | 4 | 0 | 2 | 0 | 1 | -2 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 0 | -1 | 0 | -1 | orthogonal lifted from S3wrC2 |
| ρ21 | 4 | 4 | 2 | 0 | 0 | -2 | 1 | -4 | -4 | -2 | 0 | 0 | -2 | 1 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | 1 | 0 | 1 | 0 | orthogonal lifted from C2xS3wrC2 |
| ρ22 | 4 | 4 | 0 | -2 | 0 | 1 | -2 | 4 | 4 | 0 | -2 | 0 | 1 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 0 | 1 | 0 | 1 | orthogonal lifted from S3wrC2 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | -4i | 4i | 0 | 0 | 0 | 2 | -1 | 0 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 2i | -i | i | -2i | √3 | 0 | -√3 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 1 | -2 | 4i | -4i | 0 | 0 | 0 | -1 | 2 | √-3 | 0 | 0 | -√-3 | 0 | 0 | 0 | 0 | i | -2i | 2i | -i | 0 | -√3 | 0 | √3 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 1 | -2 | 4i | -4i | 0 | 0 | 0 | -1 | 2 | -√-3 | 0 | 0 | √-3 | 0 | 0 | 0 | 0 | i | -2i | 2i | -i | 0 | √3 | 0 | -√3 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 1 | -2 | -4i | 4i | 0 | 0 | 0 | -1 | 2 | -√-3 | 0 | 0 | √-3 | 0 | 0 | 0 | 0 | -i | 2i | -2i | i | 0 | -√3 | 0 | √3 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 1 | -2 | -4i | 4i | 0 | 0 | 0 | -1 | 2 | √-3 | 0 | 0 | -√-3 | 0 | 0 | 0 | 0 | -i | 2i | -2i | i | 0 | √3 | 0 | -√3 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | -4i | 4i | 0 | 0 | 0 | 2 | -1 | 0 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 2i | -i | i | -2i | -√3 | 0 | √3 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 4i | -4i | 0 | 0 | 0 | 2 | -1 | 0 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | -2i | i | -i | 2i | √3 | 0 | -√3 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | 1 | 4i | -4i | 0 | 0 | 0 | 2 | -1 | 0 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | -2i | i | -i | 2i | -√3 | 0 | √3 | 0 | complex faithful |
►On 48 points
Generators in S48
(2 25 47)(4 41 27)(6 29 43)(8 45 31)(9 35 23)(11 17 37)(13 39 19)(15 21 33)
(1 32 46)(3 48 26)(5 28 42)(7 44 30)(10 24 36)(12 38 18)(14 20 40)(16 34 22)
(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 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)
(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 21)(18 22)(19 23)(20 24)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
G:=sub<Sym(48)| (2,25,47)(4,41,27)(6,29,43)(8,45,31)(9,35,23)(11,17,37)(13,39,19)(15,21,33), (1,32,46)(3,48,26)(5,28,42)(7,44,30)(10,24,36)(12,38,18)(14,20,40)(16,34,22), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,21)(18,22)(19,23)(20,24)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)>;
G:=Group( (2,25,47)(4,41,27)(6,29,43)(8,45,31)(9,35,23)(11,17,37)(13,39,19)(15,21,33), (1,32,46)(3,48,26)(5,28,42)(7,44,30)(10,24,36)(12,38,18)(14,20,40)(16,34,22), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44), (9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,21)(18,22)(19,23)(20,24)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46) );
G=PermutationGroup([[(2,25,47),(4,41,27),(6,29,43),(8,45,31),(9,35,23),(11,17,37),(13,39,19),(15,21,33)], [(1,32,46),(3,48,26),(5,28,42),(7,44,30),(10,24,36),(12,38,18),(14,20,40),(16,34,22)], [(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)], [(1,18),(2,17),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,27),(10,26),(11,25),(12,32),(13,31),(14,30),(15,29),(16,28),(33,43),(34,42),(35,41),(36,48),(37,47),(38,46),(39,45),(40,44)], [(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,21),(18,22),(19,23),(20,24),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)]])
Matrix representation of C32:D8:5C2 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 1 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 1 | 0 |
| 0 | 72 | 0 | 1 |
| 7 | 7 | 0 | 52 |
| 14 | 14 | 52 | 52 |
| 37 | 50 | 66 | 59 |
| 7 | 37 | 66 | 59 |
| 43 | 60 | 0 | 0 |
| 13 | 30 | 0 | 0 |
| 36 | 23 | 7 | 14 |
| 43 | 30 | 7 | 66 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 25 | 25 | 72 | 0 |
| 24 | 24 | 1 | 1 |
G:=sub<GL(4,GF(73))| [1,0,1,0,0,1,1,0,0,0,72,1,0,0,72,0],[0,1,0,0,72,72,72,72,0,0,1,0,0,0,0,1],[7,14,37,7,7,14,50,37,0,52,66,66,52,52,59,59],[43,13,36,43,60,30,23,30,0,0,7,7,0,0,14,66],[0,1,25,24,1,0,25,24,0,0,72,1,0,0,0,1] >;
C32:D8:5C2 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_8\rtimes_5C_2
% in TeX
G:=Group("C3^2:D8:5C2"); // GroupNames label
G:=SmallGroup(288,871);
// by ID
G=gap.SmallGroup(288,871);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,100,675,346,80,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=e*a*e=c*b*c^-1=a^-1,b*d=d*b,e*b*e=b^-1,d*c*d=c^-1,c*e=e*c,e*d*e=c^4*d>;
// generators/relations
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