p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.51C22, C23.21C23, C22.50C24, C2.132- 1+4, C4⋊Q8⋊17C2, (C4×Q8)⋊16C2, (C4×D4).13C2, C22⋊Q8⋊18C2, C42⋊2C2⋊7C2, C4.45(C4○D4), C4⋊C4.75C22, (C2×C4).33C23, C4.4D4.8C2, C42⋊C2⋊18C2, (C2×D4).71C22, C22⋊C4.7C22, (C2×Q8).65C22, (C22×C4).75C22, C2.29(C2×C4○D4), SmallGroup(64,237)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.50C24
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=ba=ab, e2=f2=a, dcd-1=ac=ca, fdf-1=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 106 conjugacy classes, 75 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C22.50C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C22.50C24
Character table of C22.50C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 0 | 2 | 2i | 2i | -2 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2i | -2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2i | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2i | -2i | -2 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2i | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2i | 2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 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 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(2 6)(4 8)(9 29)(10 12)(11 31)(14 24)(16 22)(17 25)(18 20)(19 27)(26 28)(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 9 5 31)(2 10 6 32)(3 11 7 29)(4 12 8 30)(13 27 23 17)(14 28 24 18)(15 25 21 19)(16 26 22 20)
(1 21 5 15)(2 16 6 22)(3 23 7 13)(4 14 8 24)(9 19 31 25)(10 26 32 20)(11 17 29 27)(12 28 30 18)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (2,6)(4,8)(9,29)(10,12)(11,31)(14,24)(16,22)(17,25)(18,20)(19,27)(26,28)(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), (1,9,5,31)(2,10,6,32)(3,11,7,29)(4,12,8,30)(13,27,23,17)(14,28,24,18)(15,25,21,19)(16,26,22,20), (1,21,5,15)(2,16,6,22)(3,23,7,13)(4,14,8,24)(9,19,31,25)(10,26,32,20)(11,17,29,27)(12,28,30,18)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (2,6)(4,8)(9,29)(10,12)(11,31)(14,24)(16,22)(17,25)(18,20)(19,27)(26,28)(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), (1,9,5,31)(2,10,6,32)(3,11,7,29)(4,12,8,30)(13,27,23,17)(14,28,24,18)(15,25,21,19)(16,26,22,20), (1,21,5,15)(2,16,6,22)(3,23,7,13)(4,14,8,24)(9,19,31,25)(10,26,32,20)(11,17,29,27)(12,28,30,18) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,31),(10,32),(11,29),(12,30),(13,23),(14,24),(15,21),(16,22),(17,27),(18,28),(19,25),(20,26)], [(1,7),(2,8),(3,5),(4,6),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28)], [(2,6),(4,8),(9,29),(10,12),(11,31),(14,24),(16,22),(17,25),(18,20),(19,27),(26,28),(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)], [(1,9,5,31),(2,10,6,32),(3,11,7,29),(4,12,8,30),(13,27,23,17),(14,28,24,18),(15,25,21,19),(16,26,22,20)], [(1,21,5,15),(2,16,6,22),(3,23,7,13),(4,14,8,24),(9,19,31,25),(10,26,32,20),(11,17,29,27),(12,28,30,18)]])
C22.50C24 is a maximal subgroup of
C42.189C23 C42.199C23 C42.211C23 C42.213C23 C42.467C23 C42.469C23 C42.46C23 C42.48C23 C42.473C23 C42.477C23 C42.527C23 C42.528C23 C42.72C23 C42.73C23 C42.531C23 C42.532C23 C22.64C25 C22.69C25 C22.82C25 C22.84C25 C22.95C25 C22.99C25 C22.103C25 C22.105C25 C23.144C24 C22.110C25 C22.129C25 C22.130C25 C22.135C25 C22.150C25 C22.151C25 C22.153C25 C22.155C25 C22.157C25
C2p.2- 1+4: C42.489C23 C42.491C23 C42.63C23 C42.64C23 C42.494C23 C42.497C23 C42.501C23 C42.505C23 ...
C22.50C24 is a maximal quotient of
C23.229C24 C23.236C24 C23.237C24 C23.238C24 C23.255C24 C23.315C24 C23.321C24 C23.346C24 C24.271C23 C23.348C24 C24.286C23 C23.374C24 C24.295C23 C23.385C24 C24.304C23 C23.396C24 C24.308C23 C23.409C24 C23.412C24 C23.414C24 C24.311C23 C24.313C23 C23.424C24 C23.425C24 C24.315C23 C23.428C24 C23.432C24 C23.433C24 C24.332C23 C42.36Q8 C42.37Q8 C23.473C24 C24.341C23 C23.488C24 C23.494C24 C23.496C24 C42⋊23D4 C42⋊8Q8 C23.589C24 C23.600C24 C23.615C24 C23.627C24 C23.640C24 C23.645C24 C23.658C24 C23.659C24 C23.662C24 C23.667C24 C23.675C24 C23.676C24 C23.677C24 C23.683C24 C23.686C24 C23.687C24 C23.688C24 C23.689C24 C23.693C24 C23.698C24 C23.700C24 C23.705C24 C23.708C24 C23.710C24
C42.D2p: C42.166D4 C42.168D4 C42.170D4 C42.177D4 C42.179D4 C42.182D4 C42.183D4 C42.184D4 ...
C4⋊C4.D2p: C23.244C24 C23.247C24 C24.220C23 C24.267C23 C24.285C23 C23.369C24 C23.392C24 C23.616C24 ...
Matrix representation of C22.50C24 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,3] >;
C22.50C24 in GAP, Magma, Sage, TeX
C_2^2._{50}C_2^4 % in TeX
G:=Group("C2^2.50C2^4"); // GroupNames label
G:=SmallGroup(64,237);
// by ID
G=gap.SmallGroup(64,237);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86,297,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=b*a=a*b,e^2=f^2=a,d*c*d^-1=a*c=c*a,f*d*f^-1=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
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