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 |
(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