p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.2C22, C23.64C23, C2.7(C4×D4), C22⋊C4⋊5C4, (C2×C42)⋊2C2, (C2×C4).100D4, C23.7(C2×C4), C2.3(C4⋊D4), C22.37(C2×D4), C2.C42⋊3C2, C2.2(C4.4D4), C2.3(C42⋊2C2), C22.22(C4○D4), C22.37(C22×C4), (C22×C4).23C22, C2.10(C42⋊C2), C2.5(C22.D4), (C2×C4⋊C4)⋊3C2, (C2×C4).17(C2×C4), (C2×C22⋊C4).6C2, SmallGroup(64,69)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.C22
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=d, f2=b, eae-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, de=ed, df=fd >
Subgroups: 161 in 95 conjugacy classes, 45 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C24.C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22
Character table of C24.C22
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | i | -1 | 1 | -1 | -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 | i | 1 | -1 | 1 | -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 | -i | -1 | 1 | -1 | 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 | -i | 1 | -1 | 1 | 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 | -i | 1 | -1 | 1 | 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 | -i | -1 | 1 | -1 | 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 | i | 1 | -1 | 1 | -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 | i | -1 | 1 | -1 | -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 | -2 | 0 | 0 | 0 | -2 | 2 | 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 | 0 | 0 | 0 | 2 | -2 | -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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 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 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -2i | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 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 |
| ρ24 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 2i | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 |
| ρ26 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -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 | 2i | 0 | -2i | -2i | 2i | 0 | 0 | 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 | -2i | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(2 8)(4 6)(9 20)(10 32)(11 18)(12 30)(13 28)(15 26)(17 21)(19 23)(22 29)(24 31)
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 28)(14 25)(15 26)(16 27)(17 32)(18 29)(19 30)(20 31)
(1 25)(2 26)(3 27)(4 28)(5 16)(6 13)(7 14)(8 15)(9 20)(10 17)(11 18)(12 19)(21 32)(22 29)(23 30)(24 31)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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 18 7 29)(2 12 8 23)(3 20 5 31)(4 10 6 21)(9 16 24 27)(11 14 22 25)(13 32 28 17)(15 30 26 19)
G:=sub<Sym(32)| (2,8)(4,6)(9,20)(10,32)(11,18)(12,30)(13,28)(15,26)(17,21)(19,23)(22,29)(24,31), (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,28)(14,25)(15,26)(16,27)(17,32)(18,29)(19,30)(20,31), (1,25)(2,26)(3,27)(4,28)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19)(21,32)(22,29)(23,30)(24,31), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,18,7,29)(2,12,8,23)(3,20,5,31)(4,10,6,21)(9,16,24,27)(11,14,22,25)(13,32,28,17)(15,30,26,19)>;
G:=Group( (2,8)(4,6)(9,20)(10,32)(11,18)(12,30)(13,28)(15,26)(17,21)(19,23)(22,29)(24,31), (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,28)(14,25)(15,26)(16,27)(17,32)(18,29)(19,30)(20,31), (1,25)(2,26)(3,27)(4,28)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19)(21,32)(22,29)(23,30)(24,31), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,18,7,29)(2,12,8,23)(3,20,5,31)(4,10,6,21)(9,16,24,27)(11,14,22,25)(13,32,28,17)(15,30,26,19) );
G=PermutationGroup([[(2,8),(4,6),(9,20),(10,32),(11,18),(12,30),(13,28),(15,26),(17,21),(19,23),(22,29),(24,31)], [(1,7),(2,8),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,28),(14,25),(15,26),(16,27),(17,32),(18,29),(19,30),(20,31)], [(1,25),(2,26),(3,27),(4,28),(5,16),(6,13),(7,14),(8,15),(9,20),(10,17),(11,18),(12,19),(21,32),(22,29),(23,30),(24,31)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,18,7,29),(2,12,8,23),(3,20,5,31),(4,10,6,21),(9,16,24,27),(11,14,22,25),(13,32,28,17),(15,30,26,19)]])
C24.C22 is a maximal subgroup of
C43⋊2C2 C4×C4⋊D4 C4×C22.D4 C4×C42⋊2C2 C23.194C24 C24.192C23 C24.547C23 C23.201C24 C23.203C24 C24.195C23 C24.198C23 C23.214C24 C23.215C24 C24.203C23 C24.204C23 C24.205C23 C23.224C24 C24.208C23 C23.229C24 C23.235C24 C24.212C23 C23.240C24 C23.241C24 C24.215C23 C24.218C23 C24.221C23 C23.255C24 C24.223C23 C23.257C24 C24.225C23 C23.259C24 C24.227C23 C23.261C24 C24.230C23 C23.311C24 C23.313C24 C24.249C23 C23.315C24 C23.318C24 C23.321C24 C23.323C24 C23.324C24 C24.258C23 C24.259C23 C23.329C24 C24.269C23 C23.344C24 C23.348C24 C24.276C23 C23.356C24 C24.278C23 C24.279C23 C23.359C24 C23.360C24 C24.282C23 C24.283C23 C23.364C24 C24.285C23 C24.286C23 C23.367C24 C23.368C24 C23.369C24 C24.289C23 C24.290C23 C23.372C24 C23.374C24 C23.375C24 C24.293C23 C23.377C24 C24.295C23 C23.379C24 C23.380C24 C24.573C23 C23.382C24 C23.385C24 C23.388C24 C24.301C23 C23.390C24 C23.391C24 C23.392C24 C24.304C23 C23.395C24 C23.397C24 C23.398C24 C23.400C24 C23.410C24 C23.412C24 C23.413C24 C24.309C23 C23.416C24 C23.418C24 C23.419C24 C24.311C23 C23.422C24 C23.425C24 C23.426C24 C24.315C23 C23.429C24 C23.430C24 C23.431C24 C23.432C24 C23.434C24 C42.165D4 C23.443C24 C42.168D4 C24.326C23 C24.327C23 C23.455C24 C23.456C24 C23.457C24 C23.458C24 C24.331C23 C24.332C23 C23.461C24 C42.172D4 C23.472C24 C23.473C24 C24.338C23 C24.339C23 C24.340C23 C24.341C23 C23.478C24 C24.345C23 C24.346C23 C23.491C24 C42.182D4 C23.493C24 C23.494C24 C24.347C23 C23.496C24 C24.348C23 C42⋊22D4 C42.183D4 C23.500C24 C42⋊23D4 C23.502C24 C42⋊24D4 C42.184D4 C42⋊25D4 C42⋊26D4 C42.185D4 C23.530C24 C42.189D4 C23.535C24 C42.192D4 C23.548C24 C24.375C23 C23.550C24 C23.551C24 C23.553C24 C23.554C24 C24.377C23 C42⋊32D4 C24.378C23 C42.198D4 C23.569C24 C23.570C24 C23.572C24 C23.573C24 C23.574C24 C24.385C23 C23.578C24 C23.580C24 C23.581C24 C23.584C24 C23.585C24 C24.393C23 C24.394C23 C23.589C24 C23.590C24 C24.401C23 C23.595C24 C24.403C23 C23.597C24 C24.405C23 C24.406C23 C23.600C24 C24.407C23 C23.602C24 C23.603C24 C24.408C23 C23.605C24 C23.606C24 C23.607C24 C24.412C23 C23.611C24 C23.612C24 C24.413C23 C23.615C24 C23.616C24 C23.618C24 C23.620C24 C23.621C24 C24.418C23 C23.625C24 C23.627C24 C24.420C23 C24.421C23 C23.630C24 C23.631C24 C23.633C24 C23.635C24 C23.636C24 C23.637C24 C24.426C23 C24.427C23 C23.640C24 C23.641C24 C24.428C23 C23.643C24 C24.430C23 C23.645C24 C24.432C23 C23.647C24 C23.649C24 C24.435C23 C23.651C24 C23.652C24 C24.437C23 C23.654C24 C23.656C24 C24.438C23 C23.659C24 C24.440C23 C23.663C24 C23.664C24 C24.443C23 C23.668C24 C24.445C23 C23.671C24 C23.672C24 C23.673C24 C23.675C24 C23.677C24 C23.678C24 C23.679C24 C23.681C24 C23.682C24 C23.683C24 C23.685C24 C23.686C24 C23.687C24 C23.688C24 C24.454C23 C23.693C24 C23.695C24 C23.696C24 C23.697C24 C23.698C24 C23.700C24 C23.701C24 C23.703C24 C23.707C24 C23.708C24 C23.728C24 C23.729C24 C23.731C24 C23.732C24 C23.736C24 C23.737C24 C23.738C24 C42⋊46D4 C42⋊43D4 C23.753C24 C24.598C23 C43⋊13C2 C43⋊14C2 C43⋊4C2 C43⋊5C2
C2p.(C4×D4): C42⋊42D4 C43⋊9C2 C4×C4.4D4 C42.159D4 C42⋊13D4 C42.160D4 C23.234C24 C23.236C24 ...
C24.C22 is a maximal quotient of
C24.624C23 C24.626C23 C23⋊2C42 C24.632C23 C24.633C23 C22⋊C4⋊4C8 C23.9M4(2) C4.68(C4×D4) C2.(C4×Q16) C4.10D4⋊3C4 C4.D4⋊3C4 C42.427D4 C2.(C8⋊8D4) C2.(C8⋊D4)
C24.D2p: C24.5Q8 C24.52D4 C24.14D6 C24.15D6 C24.19D6 C24.3D10 C24.4D10 C24.8D10 ...
D2p⋊C4⋊C4: D4⋊C4⋊C4 C4.67(C4×D4) M4(2).24D4 C2.(C8⋊7D4) C2.(C8⋊2D4) C42.428D4 C42.107D4 D6⋊C4⋊5C4 ...
Matrix representation of C24.C22 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 4 |
| 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 |
| 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 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,1,0,0,0,0,4,0,0,0,0,0,1,2,0,0,0,0,4],[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],[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],[2,0,0,0,0,0,3,0,0,0,0,4,2,0,0,0,0,0,1,2,0,0,0,0,4],[1,0,0,0,0,0,1,1,0,0,0,3,4,0,0,0,0,0,1,0,0,0,0,4,4] >;
C24.C22 in GAP, Magma, Sage, TeX
C_2^4.C_2^2
% in TeX
G:=Group("C2^4.C2^2"); // GroupNames label
G:=SmallGroup(64,69);
// by ID
G=gap.SmallGroup(64,69);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362,50]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d>;
// generators/relations
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