p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22⋊C4.C8, (C2×C8).18D4, C23.2(C2×C8), (C22×C8).8C4, C23.C8.3C2, (C2×C4).3M4(2), C4.42(C23⋊C4), C42⋊C2.2C4, C4.14(C4.10D4), C22.21(C22⋊C8), C42.6C22.9C2, (C2×M4(2)).146C22, C2.9(C22.M4(2)), (C22×C4).63(C2×C4), (C2×C4).345(C22⋊C4), SmallGroup(128,60)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22⋊C4.C8
G = < a,b,c,d | a2=b2=c4=1, d8=b, cac-1=ab=ba, dad-1=abc2, bc=cb, bd=db, dcd-1=abc >
Character table of C22⋊C4.C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | -i | i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -i | i | i | -i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | i | -i | ζ8 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ87 | ζ85 | linear of order 8 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | -i | i | ζ8 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | i | -i | ζ85 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ83 | ζ8 | linear of order 8 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | -i | i | ζ83 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ85 | ζ87 | linear of order 8 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | -i | i | ζ85 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | i | -i | ζ83 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | i | -i | ζ87 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | -i | i | ζ87 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ8 | ζ83 | linear of order 8 |
| ρ17 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | -4 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 2ζ87 | 0 | 0 | 0 | 2ζ8 | 0 | 2ζ83 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 2ζ85 | 0 | 0 | 0 | 2ζ83 | 0 | 2ζ8 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 2ζ83 | 0 | 0 | 0 | 2ζ85 | 0 | 2ζ87 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 2ζ8 | 0 | 0 | 0 | 2ζ87 | 0 | 2ζ85 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 32 points
Generators in S32
(1 17)(2 18)(3 27)(4 28)(5 21)(6 22)(7 31)(8 32)(9 25)(10 26)(11 19)(12 20)(13 29)(14 30)(15 23)(16 24)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 5 9 13)(2 22)(3 7 11 15)(4 32)(6 26)(8 20)(10 30)(12 24)(14 18)(16 28)(17 29 25 21)(19 31 27 23)
(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)
G:=sub<Sym(32)| (1,17)(2,18)(3,27)(4,28)(5,21)(6,22)(7,31)(8,32)(9,25)(10,26)(11,19)(12,20)(13,29)(14,30)(15,23)(16,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,5,9,13)(2,22)(3,7,11,15)(4,32)(6,26)(8,20)(10,30)(12,24)(14,18)(16,28)(17,29,25,21)(19,31,27,23), (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)>;
G:=Group( (1,17)(2,18)(3,27)(4,28)(5,21)(6,22)(7,31)(8,32)(9,25)(10,26)(11,19)(12,20)(13,29)(14,30)(15,23)(16,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,5,9,13)(2,22)(3,7,11,15)(4,32)(6,26)(8,20)(10,30)(12,24)(14,18)(16,28)(17,29,25,21)(19,31,27,23), (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) );
G=PermutationGroup([(1,17),(2,18),(3,27),(4,28),(5,21),(6,22),(7,31),(8,32),(9,25),(10,26),(11,19),(12,20),(13,29),(14,30),(15,23),(16,24)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,5,9,13),(2,22),(3,7,11,15),(4,32),(6,26),(8,20),(10,30),(12,24),(14,18),(16,28),(17,29,25,21),(19,31,27,23)], [(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)])
Matrix representation of C22⋊C4.C8 ►in GL4(𝔽17) generated by
| 1 | 2 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,2,16,0,0,0,0,16,0,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,4,0,0,0,4,0,0,0,0,13,4,0,0,9,4],[0,0,9,8,0,0,0,8,1,0,0,0,0,1,0,0] >;
C22⋊C4.C8 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4.C_8
% in TeX
G:=Group("C2^2:C4.C8"); // GroupNames label
G:=SmallGroup(128,60);
// by ID
G=gap.SmallGroup(128,60);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,346,521,136,2804,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=1,d^8=b,c*a*c^-1=a*b=b*a,d*a*d^-1=a*b*c^2,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c>;
// generators/relations
Export
Subgroup lattice of C22⋊C4.C8 in TeX
Character table of C22⋊C4.C8 in TeX