p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.46C24, C23.47C23, C42.47C22, C2.122- 1+4, (C4×Q8)⋊14C2, (C4×D4).11C2, C22⋊Q8⋊16C2, C42.C2⋊8C2, C42⋊2C2⋊5C2, C4.35(C4○D4), C4⋊C4.35C22, (C2×C4).56C23, C42⋊C2⋊15C2, (C2×D4).70C22, (C2×Q8).64C22, C22.10(C4○D4), C22⋊C4.30C22, (C22×C4).72C22, C22.D4.2C2, (C2×C4⋊C4)⋊21C2, C2.25(C2×C4○D4), SmallGroup(64,233)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.46C24
G = < a,b,c,d,e,f | a2=b2=f2=1, c2=e2=a, d2=b, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >
Subgroups: 141 in 107 conjugacy classes, 75 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C22.46C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C22.46C24
Character table of C22.46C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2 | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2 | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 2 | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | 2i | 0 | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 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 30)(2 31)(3 32)(4 29)(5 20)(6 17)(7 18)(8 19)(9 21)(10 22)(11 23)(12 24)(13 27)(14 28)(15 25)(16 26)
(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 22 30 10)(2 21 31 9)(3 24 32 12)(4 23 29 11)(5 28 20 14)(6 27 17 13)(7 26 18 16)(8 25 19 15)
(1 5 3 7)(2 17 4 19)(6 29 8 31)(9 27 11 25)(10 14 12 16)(13 23 15 21)(18 30 20 32)(22 28 24 26)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 18)(10 19)(11 20)(12 17)(25 32)(26 29)(27 30)(28 31)
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), (1,30)(2,31)(3,32)(4,29)(5,20)(6,17)(7,18)(8,19)(9,21)(10,22)(11,23)(12,24)(13,27)(14,28)(15,25)(16,26), (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), (1,22,30,10)(2,21,31,9)(3,24,32,12)(4,23,29,11)(5,28,20,14)(6,27,17,13)(7,26,18,16)(8,25,19,15), (1,5,3,7)(2,17,4,19)(6,29,8,31)(9,27,11,25)(10,14,12,16)(13,23,15,21)(18,30,20,32)(22,28,24,26), (1,13)(2,14)(3,15)(4,16)(5,23)(6,24)(7,21)(8,22)(9,18)(10,19)(11,20)(12,17)(25,32)(26,29)(27,30)(28,31)>;
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), (1,30)(2,31)(3,32)(4,29)(5,20)(6,17)(7,18)(8,19)(9,21)(10,22)(11,23)(12,24)(13,27)(14,28)(15,25)(16,26), (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), (1,22,30,10)(2,21,31,9)(3,24,32,12)(4,23,29,11)(5,28,20,14)(6,27,17,13)(7,26,18,16)(8,25,19,15), (1,5,3,7)(2,17,4,19)(6,29,8,31)(9,27,11,25)(10,14,12,16)(13,23,15,21)(18,30,20,32)(22,28,24,26), (1,13)(2,14)(3,15)(4,16)(5,23)(6,24)(7,21)(8,22)(9,18)(10,19)(11,20)(12,17)(25,32)(26,29)(27,30)(28,31) );
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)], [(1,30),(2,31),(3,32),(4,29),(5,20),(6,17),(7,18),(8,19),(9,21),(10,22),(11,23),(12,24),(13,27),(14,28),(15,25),(16,26)], [(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)], [(1,22,30,10),(2,21,31,9),(3,24,32,12),(4,23,29,11),(5,28,20,14),(6,27,17,13),(7,26,18,16),(8,25,19,15)], [(1,5,3,7),(2,17,4,19),(6,29,8,31),(9,27,11,25),(10,14,12,16),(13,23,15,21),(18,30,20,32),(22,28,24,26)], [(1,13),(2,14),(3,15),(4,16),(5,23),(6,24),(7,21),(8,22),(9,18),(10,19),(11,20),(12,17),(25,32),(26,29),(27,30),(28,31)]])
C22.46C24 is a maximal subgroup of
C42.461C23 C42.465C23 C42.45C23 C42.47C23 C42.50C23 C42.52C23 C42.472C23 C42.476C23 C22.64C25 C22.80C25 C22.81C25 C22.82C25 C22.84C25 C22.94C25 C22.101C25 C22.102C25 C22.105C25 C23.144C24 C22.110C25 C22.113C25 C22.122C25 C22.124C25 C22.127C25 C22.128C25 C22.130C25 C22.131C25 C22.140C25 C22.142C25 C22.153C25 C22.155C25 C22.156C25
C2p.2- 1+4: C42.485C23 C42.486C23 C42.57C23 C42.58C23 C42.62C23 C42.63C23 C42.492C23 C42.493C23 ...
C22.46C24 is a maximal quotient of
C23.225C24 C23.226C24 C24.208C23 C23.234C24 C23.238C24 C23.241C24 C23.252C24 C23.253C24 C23.255C24 C23.313C24 C23.315C24 C24.252C23 C24.563C23 C23.321C24 C23.323C24 C24.567C23 C23.368C24 C24.289C23 C23.377C24 C24.295C23 C23.379C24 C24.573C23 C24.576C23 C23.385C24 C24.299C23 C23.388C24 C24.577C23 C24.304C23 C23.395C24 C23.396C24 C23.398C24 C24.308C23 C24.579C23 C23.408C24 C23.411C24 C23.414C24 C24.309C23 C23.417C24 C23.420C24 C23.424C24 C23.425C24 C24.315C23 C23.428C24 C23.429C24 C23.430C24 C23.432C24 C23.433C24 C24.326C23 C42.36Q8 C23.473C24 C24.339C23 C24.341C23 C23.485C24 C23.490C24 C23.496C24 C42⋊23D4 C42.38Q8 C42⋊25D4 C42⋊9Q8 C24.394C23 C23.595C24 C24.405C23 C23.602C24 C24.426C23 C24.427C23 C23.641C24 C23.643C24 C24.430C23 C23.645C24 C23.647C24 C24.435C23 C23.651C24 C23.654C24 C23.658C24 C24.440C23 C23.662C24 C23.664C24 C24.443C23 C23.666C24 C23.667C24 C23.668C24 C23.669C24 C24.445C23 C23.671C24 C23.672C24 C23.673C24 C23.674C24 C23.675C24 C23.676C24 C23.677C24 C23.679C24 C23.681C24 C23.687C24 C23.691C24 C23.693C24 C23.694C24
C42.D2p: C42.165D4 C42.172D4 C42.174D4 C42.183D4 C42.184D4 C42.185D4 C42.94D6 C42.96D6 ...
C4⋊C4.D2p: C24.558C23 C23.244C24 C24.268C23 C24.569C23 C24.279C23 C23.360C24 C23.362C24 C24.572C23 ...
Matrix representation of C22.46C24 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,0,2,0,0,3,0],[1,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2],[2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4] >;
C22.46C24 in GAP, Magma, Sage, TeX
C_2^2._{46}C_2^4 % in TeX
G:=Group("C2^2.46C2^4"); // GroupNames label
G:=SmallGroup(64,233);
// by ID
G=gap.SmallGroup(64,233);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,86,297]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=f^2=1,c^2=e^2=a,d^2=b,a*b=b*a,d*c*d^-1=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,c*f=f*c,d*e=e*d,e*f=f*e>;
// generators/relations
Export