p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.3C22, C23.66C23, (C2×C4)⋊6D4, (C2×D4)⋊4C4, C2.9(C4×D4), (C2×C42)⋊5C2, C4⋊1(C22⋊C4), C23.8(C2×C4), C2.2(C4⋊1D4), C2.5(C4⋊D4), (C22×D4).2C2, C22.39(C2×D4), C2.3(C4.4D4), (C22×C4).5C22, C22.24(C4○D4), C22.39(C22×C4), (C2×C4⋊C4)⋊4C2, (C2×C22⋊C4)⋊3C2, (C2×C4).41(C2×C4), C2.8(C2×C22⋊C4), SmallGroup(64,71)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.3C22
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=d, f2=c, 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: 233 in 129 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C24.3C22
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22
Character table of C24.3C22
| 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 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -1 | 1 | -1 | 1 | -i | i | i | -i | i | -i | -i | i | i | -i | -i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | 1 | -1 | 1 | -1 | i | -i | i | -i | i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -1 | 1 | -1 | 1 | -i | i | i | -i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | 1 | -1 | 1 | -1 | i | -i | i | -i | i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -1 | 1 | -1 | 1 | i | -i | -i | i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | i | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -1 | 1 | -1 | 1 | i | -i | -i | i | -i | i | i | -i | -i | i | i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 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 | 0 | -2 | -2 | 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 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 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 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -2i | 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 | 0 | 2i | 2i | -2i | 0 | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 14)(2 17)(3 16)(4 19)(5 30)(6 28)(7 32)(8 26)(9 13)(10 20)(11 15)(12 18)(21 29)(22 27)(23 31)(24 25)
(1 10)(2 11)(3 12)(4 9)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(13 24)(14 21)(15 22)(16 23)
(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 21 29 14)(2 15 30 22)(3 23 31 16)(4 13 32 24)(5 11 17 27)(6 28 18 12)(7 9 19 25)(8 26 20 10)
G:=sub<Sym(32)| (1,14)(2,17)(3,16)(4,19)(5,30)(6,28)(7,32)(8,26)(9,13)(10,20)(11,15)(12,18)(21,29)(22,27)(23,31)(24,25), (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,25)(10,26)(11,27)(12,28)(13,24)(14,21)(15,22)(16,23), (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), (1,21,29,14)(2,15,30,22)(3,23,31,16)(4,13,32,24)(5,11,17,27)(6,28,18,12)(7,9,19,25)(8,26,20,10)>;
G:=Group( (1,14)(2,17)(3,16)(4,19)(5,30)(6,28)(7,32)(8,26)(9,13)(10,20)(11,15)(12,18)(21,29)(22,27)(23,31)(24,25), (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,25)(10,26)(11,27)(12,28)(13,24)(14,21)(15,22)(16,23), (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), (1,21,29,14)(2,15,30,22)(3,23,31,16)(4,13,32,24)(5,11,17,27)(6,28,18,12)(7,9,19,25)(8,26,20,10) );
G=PermutationGroup([[(1,14),(2,17),(3,16),(4,19),(5,30),(6,28),(7,32),(8,26),(9,13),(10,20),(11,15),(12,18),(21,29),(22,27),(23,31),(24,25)], [(1,10),(2,11),(3,12),(4,9),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18),(25,32),(26,29),(27,30),(28,31)], [(1,29),(2,30),(3,31),(4,32),(5,17),(6,18),(7,19),(8,20),(9,25),(10,26),(11,27),(12,28),(13,24),(14,21),(15,22),(16,23)], [(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)], [(1,21,29,14),(2,15,30,22),(3,23,31,16),(4,13,32,24),(5,11,17,27),(6,28,18,12),(7,9,19,25),(8,26,20,10)]])
C24.3C22 is a maximal subgroup of
(C2×C4).98D8 (C2×D4)⋊C8 (C2×SD16)⋊15C4 C8⋊C22⋊C4 D4⋊C4⋊C4 C4.67(C4×D4) M4(2)⋊D4 (C2×D4)⋊Q8 C4⋊C4.84D4 (C2×D4)⋊2Q8 (C2×C4)⋊5SD16 (C2×C4).24D8 C42⋊8C4⋊C2 (C2×C4).27D8 C43⋊9C2 C23.179C24 C4×C4⋊D4 C4×C4.4D4 C4×C4⋊1D4 C23.191C24 C24.542C23 C23.201C24 C23.203C24 C42⋊13D4 C42.160D4 C42⋊14D4 C23.215C24 C24.204C23 D4×C22⋊C4 C23.223C24 C23.236C24 C24.212C23 C23.240C24 C24.215C23 C24.217C23 C24.219C23 C24.220C23 C24.223C23 C23.257C24 C24.225C23 C23.259C24 C24.244C23 C23.308C24 C23.311C24 C24.249C23 C23.316C24 C24.254C23 C23.322C24 C23.324C24 C24.259C23 C23.328C24 C24.263C23 C24.264C23 C23.333C24 C23.335C24 C23.345C24 C24.271C23 C23.348C24 C23.352C24 C23.354C24 C23.356C24 C24.279C23 C24.282C23 C23.364C24 C23.367C24 C24.290C23 C23.372C24 C24.293C23 C23.379C24 C23.390C24 C23.391C24 C23.400C24 C23.401C24 C23.404C24 C23.412C24 C23.413C24 C23.416C24 C23.426C24 C23.431C24 C42.166D4 C23.439C24 C42⋊19D4 C42⋊20D4 C42.167D4 C42⋊21D4 C42.168D4 C42.171D4 C24.327C23 C23.455C24 C23.457C24 C23.458C24 C24.331C23 C24.332C23 C42.173D4 C42.175D4 C42.178D4 C42.182D4 C23.493C24 C24.347C23 C24.348C23 C42⋊23D4 C23.502C24 C42⋊24D4 C42⋊26D4 C42⋊27D4 C42⋊28D4 C23.524C24 C42.193D4 C42.194D4 C23.544C24 C23.548C24 C23.551C24 C24.377C23 C42⋊32D4 C23.570C24 C23.576C24 C23.578C24 C25⋊C22 C23.581C24 C24.389C23 C23.583C24 C23.585C24 C23.591C24 C23.592C24 C24.401C23 C24.403C23 C23.597C24 C24.406C23 C24.407C23 C23.603C24 C23.605C24 C23.606C24 C23.607C24 C24.412C23 C23.611C24 C24.413C23 C24.418C23 C23.624C24 C23.627C24 C23.630C24 C23.632C24 C23.633C24 C23.637C24 C23.640C24 C24.434C23 C23.649C24 C23.651C24 C23.652C24 C24.437C23 C23.656C24 C24.438C23 C24.440C23 C24.448C23 C24.450C23 C23.686C24 C23.696C24 C23.697C24 C23.700C24 C23.703C24 C24.456C23 C23.708C24 C23.728C24 C23.729C24 C23.730C24 C42⋊46D4 C24.598C23 C42⋊47D4 C43⋊12C2 C43⋊13C2 C43⋊14C2
C24.D2p: C24.6D4 2+ 1+4⋊3C4 C24.21D4 M4(2)⋊6D4 C24.24D6 C24.30D6 C24.13D10 C24.19D10 ...
(C2×D4p)⋊C4: (C2×C4)⋊9D8 (C2×C4)⋊6D8 (C2×D8)⋊10C4 (C2×C4)⋊6D12 (C2×D12)⋊10C4 (C2×C4)⋊6D20 (C2×D20)⋊22C4 C2.(D4×F5) ...
C2.(C8⋊pD4): C2.(C8⋊7D4) C2.(C8⋊2D4) (C2×C4)⋊9SD16 C8⋊(C22⋊C4) (C2×C4)⋊2D8 (C22×D8).C2 (C2×C4)⋊3SD16 (C2×C4)⋊3D8 ...
C24.3C22 is a maximal quotient of
C24.625C23 C23⋊2C42 C24.635C23 C42.325D4 C42.109D4 C42.431D4 C42.432D4 C42.433D4 C42.110D4 C42.111D4 C42.112D4 C43⋊C2 C42⋊8D4 C24.175C23 M4(2)⋊12D4 C42.114D4 C42.115D4 (C2×C4)⋊9SD16 (C2×C4)⋊6Q16 (C2×Q16)⋊10C4 C8⋊(C22⋊C4) M4(2).31D4 M4(2).33D4 M4(2)⋊13D4 C42.117D4 C42.118D4 C42.119D4
(C2×D4p)⋊C4: (C2×C4)⋊6D8 (C2×D8)⋊10C4 C42.326D4 C42.116D4 M4(2).30D4 M4(2).32D4 (C2×C4)⋊6D12 (C2×D12)⋊10C4 ...
C24.D2p: C24.50D4 C24.5Q8 C24.24D6 C24.30D6 C24.13D10 C24.19D10 C24.13D14 C24.19D14 ...
Matrix representation of C24.3C22 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 4 | 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 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[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],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,3,0],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0] >;
C24.3C22 in GAP, Magma, Sage, TeX
C_2^4._3C_2^2
% in TeX
G:=Group("C2^4.3C2^2"); // GroupNames label
G:=SmallGroup(64,71);
// by ID
G=gap.SmallGroup(64,71);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362,86]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c,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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