direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C2×C8.C22, C4.6C24, C8.1C23, Q16⋊3C22, D4.3C23, C23.51D4, Q8.3C23, SD16⋊2C22, M4(2)⋊4C22, C4.65(C2×D4), (C2×C4).50D4, (C2×Q16)⋊11C2, (C2×SD16)⋊5C2, (C22×Q8)⋊9C2, (C2×M4(2))⋊4C2, (C2×C8).25C22, (C2×C4).41C23, (C2×Q8)⋊15C22, C2.28(C22×D4), C22.24(C2×D4), C4○D4.12C22, (C2×D4).74C22, (C22×C4).80C22, (C2×C4○D4).12C2, SmallGroup(64,255)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8.C22
G = < a,b,c,d | a2=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 185 in 129 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×C8.C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, C2×C8.C22
Character table of C2×C8.C22
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 32 points
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)
G:=sub<Sym(32)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30)], [(1,29),(2,26),(3,31),(4,28),(5,25),(6,30),(7,27),(8,32),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17)]])
C2×C8.C22 is a maximal subgroup of
C8.C22⋊C4 M4(2).46D4 C42.6D4 M4(2).49D4 C42⋊10D4 C42.130D4 M4(2)⋊4D4 M4(2).D4 M4(2).5D4 M4(2).6D4 M4(2).9D4 M4(2).11D4 C42.276C23 C24.178D4 C42.13C23 C42.212D4 C42.445D4 C42.16C23 C42.17C23 M4(2)⋊15D4 M4(2)⋊17D4 M4(2)⋊8D4 M4(2)⋊9D4 M4(2)⋊10D4 M4(2).20D4 Q16⋊4D4 C4.C25
D4.pD4⋊C2: C24.104D4 C24.106D4 D4.(C2×D4) Q8.(C2×D4) (C2×Q8)⋊17D4 C42.446D4 C42.19C23 M4(2).38D4 ...
C2×C8.C22 is a maximal quotient of
C24.178D4 C24.106D4 C42.212D4 C42.445D4 M4(2)⋊15D4 C42.220D4 C42.448D4 C24.183D4 C24.118D4 C42.451D4 C42.226D4 C42.228D4 C42.230D4 C42.231D4 C42.234D4 C42.235D4 C42.241D4 C42.243D4 M4(2)⋊8D4 M4(2)⋊5Q8 C42.256D4 C42.258D4 C42.259D4 C42.262D4 C24.123D4 C24.126D4 C24.128D4 C24.129D4 C42.264D4 C42.267D4 C42.268D4 C42.273D4 C42.274D4 C42.276D4 C42.278D4 C42.281D4 C42.283D4 C42.288D4 C42.289D4 C42.290D4 C42.291D4 C42.296D4 C42.300D4 C42.302D4 C42.303D4 SD16⋊6D4 Q16⋊9D4 SD16⋊3D4 Q16⋊5D4 C42.47C23 C42.49C23 C42.51C23 C42.55C23 C42.477C23 C42.478C23 C42.480C23 C42.482C23 C42.58C23 C42.60C23 C42.497C23 C42.498C23 C42.510C23 C42.513C23 C42.515C23 C42.516C23 SD16⋊Q8 Q16⋊4Q8 C42.73C23 C42.75C23
Matrix representation of C2×C8.C22 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 2 | 0 | 0 | 0 | 0 |
| 9 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 12 | 5 |
| 0 | 0 | 5 | 5 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 0 | 16 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,5,5,5,5,0,0,12,5,12,5,0,0,0,0,12,12,0,0,0,0,5,12],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,16,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,2,0,1,0,0,0,0,2,0,1] >;
C2×C8.C22 in GAP, Magma, Sage, TeX
C_2\times C_8.C_2^2
% in TeX
G:=Group("C2xC8.C2^2"); // GroupNames label
G:=SmallGroup(64,255);
// by ID
G=gap.SmallGroup(64,255);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,217,199,650,1444,730,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations
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