p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.23D4, C23.62C23, C24.27C22, (C2×C4)⋊9D4, (C2×D4)⋊3C4, C2.5(C4×D4), C23⋊2(C2×C4), (C23×C4)⋊1C2, C2.3C22≀C2, C2.2(C4⋊D4), (C22×D4).1C2, C22.35(C2×D4), C22⋊1(C22⋊C4), C2.C42⋊8C2, C22.20(C4○D4), C22.35(C22×C4), (C22×C4).90C22, C2.3(C22.D4), (C2×C4)⋊3(C2×C4), (C2×C22⋊C4)⋊2C2, C2.7(C2×C22⋊C4), SmallGroup(64,67)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.23D4
G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, eae=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 249 in 143 conjugacy classes, 53 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C23.23D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4
Character table of C23.23D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -i | i | i | -i | i | -i | i | -i | i | 1 | -i | i | -i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -i | i | -i | i | i | -1 | -i | -i | i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | -i | i | -i | i | -i | i | i | -1 | -i | i | -i | 1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | i | -i | i | -i | i | 1 | -i | -i | i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | i | i | -i | i | -i | -i | -1 | i | i | -i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -i | i | -i | i | -i | 1 | i | -i | i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | -i | i | -i | i | -i | 1 | i | i | -i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | i | -i | i | -i | i | -i | -i | -1 | i | -i | i | 1 | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 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 |
| ρ20 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 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 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 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 | orthogonal lifted from D4 |
| 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 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 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 | -2i | 2i | -2i | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 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 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 31)(6 32)(7 29)(8 30)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 20)(2 17)(3 18)(4 19)(5 15)(6 16)(7 13)(8 14)(9 30)(10 31)(11 32)(12 29)(21 25)(22 26)(23 27)(24 28)
(1 12)(2 9)(3 10)(4 11)(5 23)(6 24)(7 21)(8 22)(13 25)(14 26)(15 27)(16 28)(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 10)(3 12)(5 25)(6 16)(7 27)(8 14)(13 23)(15 21)(18 29)(20 31)(22 26)(24 28)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,31)(6,32)(7,29)(8,30)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,20)(2,17)(3,18)(4,19)(5,15)(6,16)(7,13)(8,14)(9,30)(10,31)(11,32)(12,29)(21,25)(22,26)(23,27)(24,28), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,25)(14,26)(15,27)(16,28)(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), (1,10)(3,12)(5,25)(6,16)(7,27)(8,14)(13,23)(15,21)(18,29)(20,31)(22,26)(24,28)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,31)(6,32)(7,29)(8,30)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,20)(2,17)(3,18)(4,19)(5,15)(6,16)(7,13)(8,14)(9,30)(10,31)(11,32)(12,29)(21,25)(22,26)(23,27)(24,28), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,25)(14,26)(15,27)(16,28)(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), (1,10)(3,12)(5,25)(6,16)(7,27)(8,14)(13,23)(15,21)(18,29)(20,31)(22,26)(24,28) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,31),(6,32),(7,29),(8,30),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(1,20),(2,17),(3,18),(4,19),(5,15),(6,16),(7,13),(8,14),(9,30),(10,31),(11,32),(12,29),(21,25),(22,26),(23,27),(24,28)], [(1,12),(2,9),(3,10),(4,11),(5,23),(6,24),(7,21),(8,22),(13,25),(14,26),(15,27),(16,28),(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)], [(1,10),(3,12),(5,25),(6,16),(7,27),(8,14),(13,23),(15,21),(18,29),(20,31),(22,26),(24,28)]])
C23.23D4 is a maximal subgroup of
C23.179C24 C4×C22≀C2 C4×C4⋊D4 C4×C22.D4 C24.542C23 C24.547C23 C23.203C24 C24.195C23 C24.198C23 C23.215C24 C24.204C23 C24.205C23 D4×C22⋊C4 C24.549C23 C23.235C24 C23.240C24 C23.241C24 C24.215C23 C24.218C23 C24.219C23 C24.221C23 C24.223C23 C23.257C24 C24.225C23 C23.259C24 C23.261C24 C23.262C24 C24⋊7D4 C23.304C24 C24.243C23 C24.244C23 C23.309C24 C24⋊8D4 C23.311C24 C23.313C24 C23.316C24 C24.252C23 C23.318C24 C24.563C23 C24.254C23 C23.324C24 C24.258C23 C23.327C24 C23.328C24 C24.262C23 C24.263C23 C24.565C23 C24.269C23 C23.344C24 C23.345C24 C23.349C24 C23.350C24 C23.356C24 C24.278C23 C23.359C24 C24.282C23 C24.283C23 C23.364C24 C23.367C24 C24.289C23 C24.290C23 C23.372C24 C23.374C24 C24.293C23 C23.377C24 C23.379C24 C24.573C23 C24.300C23 C23.388C24 C23.398C24 C23.404C24 C23.410C24 C23.416C24 C23.418C24 C24.311C23 C23.426C24 C23.431C24 C23.434C24 C42⋊17D4 C23.443C24 C42⋊21D4 C24.326C23 C24.327C23 C23.455C24 C23.457C24 C24.331C23 C24.332C23 C24.583C23 C23.472C24 C24.340C23 C23.478C24 C23.479C24 C23.491C24 C24.347C23 C24.348C23 C42⋊22D4 C23.500C24 C42⋊23D4 C23.502C24 C42⋊24D4 C42⋊25D4 C42⋊26D4 C24⋊10D4 C24.587C23 C24.589C23 C23.530C24 C42⋊29D4 C23.535C24 C42⋊30D4 C24.592C23 C23.543C24 C23.548C24 C24.375C23 C23.553C24 C23.568C24 C23.569C24 C23.571C24 C23.572C24 C23.573C24 C23.576C24 C23.578C24 C23.581C24 C23.583C24 C23.584C24 C24.393C23 C24.394C23 C24.395C23 C23.591C24 C23.592C24 C23.593C24 C23.595C24 C24.403C23 C23.597C24 C24.406C23 C23.600C24 C24.407C23 C23.602C24 C23.603C24 C23.605C24 C23.606C24 C23.608C24 C24.411C23 C24.412C23 C23.612C24 C24.413C23 C23.615C24 C23.617C24 C24.418C23 C24.420C23 C23.630C24 C23.632C24 C23.637C24 C23.640C24 C23.641C24 C23.643C24 C24.432C23 C24.434C23 C23.649C24 C24.435C23 C23.651C24 C23.652C24 C24.437C23 C23.656C24 C24.438C23 C23.660C24 C24.440C23 C23.678C24 C23.679C24 C24.448C23 C23.681C24 C23.682C24 C24.450C23 C23.686C24 C23.696C24 C23.697C24 C24.456C23 C23.724C24 C23.725C24 C23.726C24 C23.727C24 C24⋊13D4 C42⋊46D4 C42⋊43D4 C23.753C24 C24.598C23
C24.D2p: C24.5D4 2+ 1+4⋊2C4 C24.22D4 C24.33D4 C24.90D4 C24.94D4 C24.95D4 C24.97D4 ...
C2p.(C4×D4): C42⋊42D4 C42⋊13D4 C42⋊14D4 C23.234C24 C24.217C23 (C2×C4)⋊9D12 (C2×C4)⋊9D20 (C2×F5)⋊D4 ...
C23.23D4 is a maximal quotient of
C24.626C23 C23⋊2C42 C24.631C23 C23.22M4(2) C23⋊2M4(2) M4(2).43D4 (C2×SD16)⋊15C4 M4(2).44D4 C8.C22⋊C4 C8⋊C22⋊C4 M4(2)⋊19D4 C4⋊Q8⋊15C4 C4.4D4⋊13C4 (C2×C8)⋊D4 C24.174C23 M4(2)⋊20D4 M4(2).45D4 M4(2).46D4 M4(2).47D4 C42.5D4 C42.6D4 C42.426D4 C4.(C4×D4) (C2×C8)⋊4D4 C42.7D4 M4(2)⋊21D4 M4(2).50D4 (C2×F5)⋊D4
C24.D2p: C24.17Q8 C24.50D4 C24.5Q8 C24.52D4 C24.72D4 C24.160D4 C24.73D4 C23.38D8 ...
(C2×C4)⋊D4p: (C2×C4)⋊9D8 (C2×C4)⋊9D12 (C2×C4)⋊9D20 (C2×C4)⋊9D28 ...
C2.(D4.pD4): (C2×SD16)⋊14C4 (C2×C4)⋊9Q16 M4(2).48D4 M4(2).49D4 ...
Matrix representation of C23.23D4 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[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],[3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,3,0],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1] >;
C23.23D4 in GAP, Magma, Sage, TeX
C_2^3._{23}D_4 % in TeX
G:=Group("C2^3.23D4"); // GroupNames label
G:=SmallGroup(64,67);
// by ID
G=gap.SmallGroup(64,67);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=e^2=1,e*a*e=a*b=b*a,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=c*d^-1>;
// generators/relations
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