direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22⋊Q8, C23⋊3Q8, C23.34C23, C22.17C24, C24.30C22, C4.61(C2×D4), C22⋊1(C2×Q8), C4⋊C4⋊10C22, (C2×C4).134D4, (C22×Q8)⋊3C2, (C2×Q8)⋊8C22, C2.6(C22×D4), C2.3(C22×Q8), (C2×C4).11C23, (C23×C4).10C2, C22.60(C2×D4), C22.30(C4○D4), C22⋊C4.11C22, (C22×C4).122C22, (C2×C4⋊C4)⋊15C2, C2.6(C2×C4○D4), (C2×C22⋊C4).11C2, SmallGroup(64,204)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22⋊Q8
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 225 in 161 conjugacy classes, 97 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C2×C22⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8
Character table of C2×C22⋊Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 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 | 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 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 29)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)(21 23)(22 24)
(1 21)(2 22)(3 23)(4 24)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 32)(18 29)(19 30)(20 31)
(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)
(1 20 3 18)(2 19 4 17)(5 25 7 27)(6 28 8 26)(9 16 11 14)(10 15 12 13)(21 31 23 29)(22 30 24 32)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(21,23)(22,24), (1,21)(2,22)(3,23)(4,24)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,32)(18,29)(19,30)(20,31), (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), (1,20,3,18)(2,19,4,17)(5,25,7,27)(6,28,8,26)(9,16,11,14)(10,15,12,13)(21,31,23,29)(22,30,24,32)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(21,23)(22,24), (1,21)(2,22)(3,23)(4,24)(5,10)(6,11)(7,12)(8,9)(13,27)(14,28)(15,25)(16,26)(17,32)(18,29)(19,30)(20,31), (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), (1,20,3,18)(2,19,4,17)(5,25,7,27)(6,28,8,26)(9,16,11,14)(10,15,12,13)(21,31,23,29)(22,30,24,32) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,17),(14,18),(15,19),(16,20),(25,30),(26,31),(27,32),(28,29)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29),(21,23),(22,24)], [(1,21),(2,22),(3,23),(4,24),(5,10),(6,11),(7,12),(8,9),(13,27),(14,28),(15,25),(16,26),(17,32),(18,29),(19,30),(20,31)], [(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)], [(1,20,3,18),(2,19,4,17),(5,25,7,27),(6,28,8,26),(9,16,11,14),(10,15,12,13),(21,31,23,29),(22,30,24,32)]])
C2×C22⋊Q8 is a maximal subgroup of
C24.45(C2×C4) C24.55D4 C24.57D4 C24.61D4 C24.160D4 C24.73D4 C24.135D4 C24.75D4 M4(2).45D4 C24.176C23 C23⋊3SD16 C23⋊2Q16 C24.85D4 C24.86D4 C42.159D4 C23.211C24 C23.214C24 C24.558C23 C23.244C24 C23.250C24 C24.227C23 C24.244C23 C23.309C24 C23.315C24 C24.252C23 C23.321C24 C23.323C24 C24.259C23 C23.327C24 C23.329C24 C24.264C23 C23.334C24 C23.335C24 C24⋊4Q8 C24.567C23 C24.568C23 C23.349C24 C23.350C24 C23.352C24 C24.282C23 C24.283C23 C24.285C23 C23.372C24 C23.377C24 C23.388C24 C24.301C23 C23.392C24 C24.308C23 C23.402C24 C24.579C23 C42.165D4 C42.166D4 C42⋊19D4 C42.167D4 C23.449C24 C23.456C24 C24.332C23 C23.461C24 C24.583C23 C23.483C24 C24.361C23 C42⋊28D4 C42.186D4 C23.525C24 C24⋊5Q8 C23.527C24 C24.374C23 C24.592C23 C23.559C24 C24.378C23 C23.572C24 C23.574C24 C23.576C24 C24.385C23 C23.580C24 C23.581C24 C23.583C24 C24.394C23 C23.589C24 C23.590C24 C23.592C24 C24.403C23 C24.405C23 C23.600C24 C23.602C24 C24.408C23 C23.620C24 C24.418C23 C24.420C23 C24.421C23 C23.630C24 C23.632C24 C23.714C24 C23.716C24 C24.462C23 C24⋊8Q8 C42⋊46D4 C24.599C23 C42.440D4 C24.106D4 M4(2)⋊15D4 C24.118D4 C23⋊4SD16 C23⋊3Q16 C24.123D4 C24.126D4 C24.128D4 C24.129D4 C2×D4×Q8 C22.78C25 C22.84C25 C22.90C25 C22.94C25 C23.144C24 C22.124C25 C22.125C25 C22.127C25 C22.130C25
C2×C22⋊Q8 is a maximal quotient of
C42.162D4 C23.309C24 C24.252C23 C24⋊4Q8 C24.567C23 C24.267C23 C24.568C23 C24.268C23 C24.569C23 C23.349C24 C23.350C24 C23.351C24 C23.352C24 C23.353C24 C23.354C24 C24.300C23 C42.166D4 C42.167D4 C42.173D4 C24.583C23 C42.174D4 C42.175D4 C42.176D4 C42.177D4 C23.479C24 C42.178D4 C42.179D4 C42.180D4 C24⋊5Q8 C23.527C24 C42.187D4 C42.188D4 C24⋊8Q8 C42.439D4 C24.599C23 C42.440D4 C42.447D4 C42.219D4 C42.220D4 C42.448D4 C42.449D4 C42.20C23 C42.21C23 C42.22C23 C42.23C23
Matrix representation of C2×C22⋊Q8 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,3,2,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0] >;
C2×C22⋊Q8 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes Q_8
% in TeX
G:=Group("C2xC2^2:Q8"); // GroupNames label
G:=SmallGroup(64,204);
// by ID
G=gap.SmallGroup(64,204);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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