p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42.55C22, C23.25C23, C22.56C24, C2.232+ 1+4, C2.152- 1+4, C4⋊D4⋊18C2, C22⋊Q8⋊19C2, C4.4D4⋊16C2, C4⋊C4.39C22, (C2×C4).38C23, C42.C2⋊11C2, (C2×D4).39C22, C22⋊C4.8C22, (C2×Q8).35C22, C22.D4⋊14C2, (C22×C4).76C22, SmallGroup(64,243)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.56C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e2=a, ab=ba, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, fcf=abc, ede-1=abd, ef=fe >
Subgroups: 181 in 110 conjugacy classes, 71 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C22.56C24
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C22.56C24
Character table of C22.56C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 2 |
| ρ6 | 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 | linear of order 2 |
| ρ8 | 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 | linear of order 2 |
| ρ10 | 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 | linear of order 2 |
| ρ12 | 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 | linear of order 2 |
| ρ14 | 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 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ17 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ18 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 32 points
(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 12)(2 9)(3 10)(4 11)(5 29)(6 30)(7 31)(8 32)(13 19)(14 20)(15 17)(16 18)(21 28)(22 25)(23 26)(24 27)
(1 25)(2 23)(3 27)(4 21)(5 17)(6 16)(7 19)(8 14)(9 26)(10 24)(11 28)(12 22)(13 31)(15 29)(18 30)(20 32)
(1 20)(2 13)(3 18)(4 15)(5 26)(6 22)(7 28)(8 24)(9 19)(10 16)(11 17)(12 14)(21 31)(23 29)(25 30)(27 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)
(5 29)(6 30)(7 31)(8 32)(13 15)(14 16)(17 19)(18 20)(21 26)(22 27)(23 28)(24 25)
G:=sub<Sym(32)| (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), (1,12)(2,9)(3,10)(4,11)(5,29)(6,30)(7,31)(8,32)(13,19)(14,20)(15,17)(16,18)(21,28)(22,25)(23,26)(24,27), (1,25)(2,23)(3,27)(4,21)(5,17)(6,16)(7,19)(8,14)(9,26)(10,24)(11,28)(12,22)(13,31)(15,29)(18,30)(20,32), (1,20)(2,13)(3,18)(4,15)(5,26)(6,22)(7,28)(8,24)(9,19)(10,16)(11,17)(12,14)(21,31)(23,29)(25,30)(27,32), (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), (5,29)(6,30)(7,31)(8,32)(13,15)(14,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)>;
G:=Group( (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), (1,12)(2,9)(3,10)(4,11)(5,29)(6,30)(7,31)(8,32)(13,19)(14,20)(15,17)(16,18)(21,28)(22,25)(23,26)(24,27), (1,25)(2,23)(3,27)(4,21)(5,17)(6,16)(7,19)(8,14)(9,26)(10,24)(11,28)(12,22)(13,31)(15,29)(18,30)(20,32), (1,20)(2,13)(3,18)(4,15)(5,26)(6,22)(7,28)(8,24)(9,19)(10,16)(11,17)(12,14)(21,31)(23,29)(25,30)(27,32), (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), (5,29)(6,30)(7,31)(8,32)(13,15)(14,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25) );
G=PermutationGroup([[(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)], [(1,12),(2,9),(3,10),(4,11),(5,29),(6,30),(7,31),(8,32),(13,19),(14,20),(15,17),(16,18),(21,28),(22,25),(23,26),(24,27)], [(1,25),(2,23),(3,27),(4,21),(5,17),(6,16),(7,19),(8,14),(9,26),(10,24),(11,28),(12,22),(13,31),(15,29),(18,30),(20,32)], [(1,20),(2,13),(3,18),(4,15),(5,26),(6,22),(7,28),(8,24),(9,19),(10,16),(11,17),(12,14),(21,31),(23,29),(25,30),(27,32)], [(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)], [(5,29),(6,30),(7,31),(8,32),(13,15),(14,16),(17,19),(18,20),(21,26),(22,27),(23,28),(24,25)]])
C22.56C24 is a maximal subgroup of
C42.2C23 C42.7C23 C42.8C23 C22.120C25 C22.122C25 C22.123C25 C22.124C25 C22.134C25 C22.142C25 C22.148C25 C22.154C25 C22.155C25 C22.156C25
C2p.2+ 1+4: C22.118C25 C22.125C25 C22.130C25 C22.131C25 C22.136C25 C22.137C25 C22.143C25 C22.147C25 ...
C22.56C24 is a maximal quotient of
C24.225C23 C23.259C24 C24.227C23 C23.261C24 C23.264C24 C23.571C24 C23.572C24 C23.576C24 C23.581C24 C24.389C23 C24.393C23 C24.394C23 C23.592C24 C24.403C23 C23.600C24 C24.407C23 C23.606C24 C23.608C24 C24.412C23 C23.630C24 C23.632C24 C24.427C23 C23.640C24 C23.641C24 C23.643C24 C24.430C23 C23.647C24 C24.435C23 C23.651C24 C23.654C24 C23.655C24 C24.440C23 C23.668C24 C23.672C24 C23.679C24 C23.681C24 C23.683C24 C23.686C24 C23.688C24 C23.698C24 C23.700C24 C23.702C24 C23.706C24 C23.708C24 C24.459C23 C23.714C24 C23.716C24 C42⋊34D4 C23.724C24 C23.726C24 C23.727C24 C23.729C24 C23.730C24 C23.731C24 C23.732C24 C23.736C24 C23.737C24 C23.739C24 C23.741C24 C42.40Q8
C42.D2p: C42.199D4 C42.145D6 C42.158D6 C42.145D10 C42.158D10 C42.145D14 C42.158D14 ...
C4⋊C4.D2p: C24.401C23 C24.408C23 C23.611C24 C6.492+ 1+4 C6.592+ 1+4 C6.692+ 1+4 C10.742- 1+4 C10.262- 1+4 ...
Matrix representation of C22.56C24 ►in GL8(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 1 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,4,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,4,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[1,0,4,4,0,0,0,0,3,4,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[1,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4] >;
C22.56C24 in GAP, Magma, Sage, TeX
C_2^2._{56}C_2^4 % in TeX
G:=Group("C2^2.56C2^4"); // GroupNames label
G:=SmallGroup(64,243);
// by ID
G=gap.SmallGroup(64,243);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,650,476,86,1347,297]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^2=a,a*b=b*a,d*c*d=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f=a*b*c,e*d*e^-1=a*b*d,e*f=f*e>;
// generators/relations
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