non-abelian, soluble, monomial
Aliases: C32⋊D8⋊5C2, C4.20S3≀C2, C32⋊Q16⋊5C2, (C3×C12).20D4, C32⋊1(C4○D8), D6.D6⋊1C2, C32⋊2SD16⋊7C2, C3⋊Dic3.1C23, D6⋊S3.5C22, C32⋊2C8.9C22, C32⋊2Q8.6C22, C2.7(C2×S3≀C2), C3⋊S3⋊3C8⋊7C2, (C3×C6).4(C2×D4), (C2×C3⋊S3).28D4, (C4×C3⋊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, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, C32⋊2C8, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C4×C3⋊S3, C32⋊D8, C32⋊2SD16, C32⋊Q16, C3⋊S3⋊3C8, D6.D6, C32⋊D8⋊5C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D8, S3≀C2, C2×S3≀C2, 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 C4○D8 |
| 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 C4○D8 |
| 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 C4○D8 |
| 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 C4○D8 |
| 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 S3≀C2 |
| 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 S3≀C2 |
| 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 C2×S3≀C2 |
| 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 C2×S3≀C2 |
| 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 C2×S3≀C2 |
| 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 S3≀C2 |
| 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 C2×S3≀C2 |
| 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 S3≀C2 |
| 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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