p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.8Q8, C23.35D4, C24.26C22, C23.61C23, C2.4(C4×D4), C22⋊C4⋊4C4, (C2×C4).98D4, C22⋊2(C4⋊C4), C2.2C22≀C2, (C23×C4).4C2, C23.14(C2×C4), C22.34(C2×D4), C2.2(C22⋊Q8), C22.12(C2×Q8), C2.C42⋊7C2, (C22×C4).4C22, C22.19(C4○D4), C22.34(C22×C4), C2.2(C22.D4), (C2×C4⋊C4)⋊2C2, (C2×C4)⋊2(C2×C4), C2.7(C2×C4⋊C4), (C2×C22⋊C4).5C2, SmallGroup(64,66)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.8Q8
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 185 in 117 conjugacy classes, 53 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C23.8Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8
Character table of C23.8Q8
| 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 | i | -i | i | -i | i | -i | i | -i | -i | 1 | -1 | i | -i | i | -1 | 1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | -i | -1 | 1 | i | -i | i | 1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | i | -1 | 1 | -i | i | -i | 1 | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | i | 1 | -1 | -i | i | -i | -1 | 1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | -i | i | i | -i | i | i | 1 | 1 | i | -i | -i | -1 | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | i | -i | -i | i | -i | i | -1 | -1 | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | -i | i | i | -i | i | -i | -1 | -1 | -i | i | i | 1 | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | i | -i | -i | i | -i | -i | 1 | 1 | -i | i | i | -1 | -1 | linear of order 4 |
| ρ17 | 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 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 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 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 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 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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 | 2i | 0 | -2i | 2i | 0 | 0 | 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 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 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 | -2i | 0 | 2i | -2i | 0 | 0 | 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 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 3)(2 4)(5 7)(6 8)(9 26)(10 27)(11 28)(12 25)(13 15)(14 16)(17 32)(18 29)(19 30)(20 31)(21 23)(22 24)
(1 6)(2 7)(3 8)(4 5)(9 29)(10 30)(11 31)(12 32)(13 23)(14 24)(15 21)(16 22)(17 25)(18 26)(19 27)(20 28)
(1 15)(2 16)(3 13)(4 14)(5 24)(6 21)(7 22)(8 23)(9 28)(10 25)(11 26)(12 27)(17 30)(18 31)(19 32)(20 29)
(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 26 3 28)(2 17 4 19)(5 27 7 25)(6 18 8 20)(9 15 11 13)(10 24 12 22)(14 32 16 30)(21 31 23 29)
G:=sub<Sym(32)| (1,3)(2,4)(5,7)(6,8)(9,26)(10,27)(11,28)(12,25)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24), (1,6)(2,7)(3,8)(4,5)(9,29)(10,30)(11,31)(12,32)(13,23)(14,24)(15,21)(16,22)(17,25)(18,26)(19,27)(20,28), (1,15)(2,16)(3,13)(4,14)(5,24)(6,21)(7,22)(8,23)(9,28)(10,25)(11,26)(12,27)(17,30)(18,31)(19,32)(20,29), (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), (1,26,3,28)(2,17,4,19)(5,27,7,25)(6,18,8,20)(9,15,11,13)(10,24,12,22)(14,32,16,30)(21,31,23,29)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,26)(10,27)(11,28)(12,25)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24), (1,6)(2,7)(3,8)(4,5)(9,29)(10,30)(11,31)(12,32)(13,23)(14,24)(15,21)(16,22)(17,25)(18,26)(19,27)(20,28), (1,15)(2,16)(3,13)(4,14)(5,24)(6,21)(7,22)(8,23)(9,28)(10,25)(11,26)(12,27)(17,30)(18,31)(19,32)(20,29), (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), (1,26,3,28)(2,17,4,19)(5,27,7,25)(6,18,8,20)(9,15,11,13)(10,24,12,22)(14,32,16,30)(21,31,23,29) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,26),(10,27),(11,28),(12,25),(13,15),(14,16),(17,32),(18,29),(19,30),(20,31),(21,23),(22,24)], [(1,6),(2,7),(3,8),(4,5),(9,29),(10,30),(11,31),(12,32),(13,23),(14,24),(15,21),(16,22),(17,25),(18,26),(19,27),(20,28)], [(1,15),(2,16),(3,13),(4,14),(5,24),(6,21),(7,22),(8,23),(9,28),(10,25),(11,26),(12,27),(17,30),(18,31),(19,32),(20,29)], [(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)], [(1,26,3,28),(2,17,4,19),(5,27,7,25),(6,18,8,20),(9,15,11,13),(10,24,12,22),(14,32,16,30),(21,31,23,29)]])
C23.8Q8 is a maximal subgroup of
C23.2M4(2) C24.182C23 C42⋊42D4 C23.178C24 C4×C22≀C2 C4×C22.D4 C4×C22⋊Q8 C23.199C24 C24.547C23 C23.203C24 C24.195C23 C42⋊13D4 C24.198C23 C42.160D4 C23.211C24 C24.203C23 C24.204C23 C24.205C23 C23.224C24 C23.225C24 D4×C4⋊C4 C23.240C24 C23.241C24 C24.558C23 C24.215C23 C24.220C23 C23.250C24 C23.255C24 C24.225C23 C23.259C24 C23.261C24 C24.230C23 C24.243C23 C24.244C23 C23.309C24 C24⋊8D4 C23.311C24 C23.313C24 C23.318C24 C24.563C23 C23.322C24 C23.324C24 C24.258C23 C24.262C23 C24.264C23 C23.334C24 C24⋊4Q8 C24.567C23 C24.568C23 C24.569C23 C24.269C23 C23.344C24 C24.271C23 C23.349C24 C23.350C24 C23.352C24 C23.354C24 C24.276C23 C23.356C24 C24.279C23 C23.360C24 C24.282C23 C24.283C23 C23.364C24 C24.285C23 C24.286C23 C23.368C24 C24.289C23 C24.572C23 C23.374C24 C23.375C24 C24.293C23 C23.377C24 C24.295C23 C23.379C24 C23.380C24 C24.573C23 C23.382C24 C24.576C23 C24.300C23 C24.577C23 C23.402C24 C23.405C24 C23.410C24 C24.309C23 C23.417C24 C23.422C24 C24.313C23 C24.315C23 C23.430C24 C23.449C24 C24.326C23 C23.456C24 C23.458C24 C24.331C23 C24.583C23 C42.175D4 C24.584C23 C23.473C24 C24.338C23 C24.339C23 C24.340C23 C24.341C23 C23.478C24 C23.479C24 C42.178D4 C23.483C24 C24.345C23 C24.346C23 C24.347C23 C24.348C23 C42⋊22D4 C42.183D4 C23.500C24 C42⋊23D4 C23.502C24 C42⋊24D4 C42⋊25D4 C42.185D4 C24.589C23 C24⋊5Q8 C23.527C24 C23.530C24 C42.190D4 C42⋊30D4 C24.374C23 C23.543C24 C23.546C24 C24.375C23 C24.376C23 C23.553C24 C23.571C24 C23.572C24 C23.574C24 C24.384C23 C23.576C24 C24.385C23 C23.580C24 C23.581C24 C24.389C23 C24.393C23 C24.394C23 C24.395C23 C23.589C24 C23.590C24 C23.591C24 C23.592C24 C23.593C24 C24.401C23 C23.595C24 C24.405C23 C24.407C23 C24.408C23 C23.606C24 C23.607C24 C23.608C24 C24.412C23 C23.611C24 C23.612C24 C24.413C23 C23.615C24 C23.617C24 C23.618C24 C23.620C24 C23.622C24 C24.418C23 C23.624C24 C24.420C23 C24.421C23 C23.632C24 C24.426C23 C24.427C23 C24.428C23 C23.643C24 C24.430C23 C23.645C24 C24.432C23 C23.647C24 C24.434C23 C23.649C24 C24.435C23 C23.651C24 C24.437C23 C24.438C23 C24.440C23 C23.663C24 C23.664C24 C24.443C23 C23.668C24 C24.445C23 C23.671C24 C23.678C24 C23.679C24 C24.448C23 C23.681C24 C23.682C24 C24.450C23 C23.686C24 C23.687C24 C23.688C24 C24.454C23 C23.696C24 C23.697C24 C24.456C23 C23.707C24 C23.724C24 C23.726C24 C23.727C24 C23.734C24 C23.735C24 C24⋊8Q8 C42.439D4 C42⋊43D4 C23.753C24 C24.599C23
C24.D2p: C24.22D4 C24.33D4 C24.91D4 C24.94D4 C24.95D4 C24.96D4 C24.97D4 C24.166D4 ...
D2p⋊(C4⋊C4): C23.231C24 D6⋊C4⋊C4 D6⋊C4⋊6C4 D10⋊3(C4⋊C4) D10⋊5(C4⋊C4) D10⋊(C4⋊C4) D14⋊C4⋊C4 D14⋊C4⋊6C4 ...
C23.8Q8 is a maximal quotient of
C24.626C23 C24.632C23 C24.634C23 C23.21M4(2) (C2×C8).195D4 C24.10Q8 C4.10D4⋊2C4 M4(2).40D4 C42⋊9(C2×C4) M4(2).41D4 Q8⋊(C4⋊C4) Q8⋊C4⋊C4 M4(2).42D4 (C2×D4).Q8
C24.D2p: C24.17Q8 C24.5Q8 C23.37D8 C24.159D4 C24.71D4 C24.21D4 C24.22D4 C24.57D6 ...
D2p⋊(C4⋊C4): C4≀C2⋊C4 C2.(C4×D8) D4⋊(C4⋊C4) D6⋊C4⋊C4 D6⋊C4⋊6C4 D10⋊3(C4⋊C4) D10⋊5(C4⋊C4) D10⋊(C4⋊C4) ...
Matrix representation of C23.8Q8 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 3 | 1 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,3,3],[3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,1,1] >;
C23.8Q8 in GAP, Magma, Sage, TeX
C_2^3._8Q_8
% in TeX
G:=Group("C2^3.8Q8"); // GroupNames label
G:=SmallGroup(64,66);
// by ID
G=gap.SmallGroup(64,66);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations
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