p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.54C22, C23.22C23, C22.53C24, C2.202+ 1+4, (C4×D4)⋊22C2, (C4×Q8)⋊17C2, C4⋊1D4.7C2, C4.47(C4○D4), C4.4D4⋊14C2, C4⋊C4.77C22, (C2×C4).35C23, (C2×D4).37C22, (C2×Q8).67C22, C22.D4⋊12C2, C22⋊C4.25C22, (C22×C4).15C22, C2.31(C2×C4○D4), SmallGroup(64,240)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.53C24
G = < a,b,c,d,e,f | a2=b2=1, c2=e2=b, d2=ba=ab, 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: 181 in 118 conjugacy classes, 75 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, C22.53C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C22.53C24
Character table of C22.53C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 0 | -2i | -2i | -2 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2i | 2i | -2 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 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 | orthogonal lifted from 2+ 1+4 |
►On 32 points
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)
(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 28 11 16)(2 15 12 27)(3 26 9 14)(4 13 10 25)(5 32 22 20)(6 19 23 31)(7 30 24 18)(8 17 21 29)
(1 7 3 5)(2 6 4 8)(9 22 11 24)(10 21 12 23)(13 17 15 19)(14 20 16 18)(25 29 27 31)(26 32 28 30)
(1 25 9 15)(2 26 10 16)(3 27 11 13)(4 28 12 14)(5 31 24 17)(6 32 21 18)(7 29 22 19)(8 30 23 20)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30), (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), (1,28,11,16)(2,15,12,27)(3,26,9,14)(4,13,10,25)(5,32,22,20)(6,19,23,31)(7,30,24,18)(8,17,21,29), (1,7,3,5)(2,6,4,8)(9,22,11,24)(10,21,12,23)(13,17,15,19)(14,20,16,18)(25,29,27,31)(26,32,28,30), (1,25,9,15)(2,26,10,16)(3,27,11,13)(4,28,12,14)(5,31,24,17)(6,32,21,18)(7,29,22,19)(8,30,23,20)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30), (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), (1,28,11,16)(2,15,12,27)(3,26,9,14)(4,13,10,25)(5,32,22,20)(6,19,23,31)(7,30,24,18)(8,17,21,29), (1,7,3,5)(2,6,4,8)(9,22,11,24)(10,21,12,23)(13,17,15,19)(14,20,16,18)(25,29,27,31)(26,32,28,30), (1,25,9,15)(2,26,10,16)(3,27,11,13)(4,28,12,14)(5,31,24,17)(6,32,21,18)(7,29,22,19)(8,30,23,20) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,24),(6,21),(7,22),(8,23),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30)], [(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)], [(1,28,11,16),(2,15,12,27),(3,26,9,14),(4,13,10,25),(5,32,22,20),(6,19,23,31),(7,30,24,18),(8,17,21,29)], [(1,7,3,5),(2,6,4,8),(9,22,11,24),(10,21,12,23),(13,17,15,19),(14,20,16,18),(25,29,27,31),(26,32,28,30)], [(1,25,9,15),(2,26,10,16),(3,27,11,13),(4,28,12,14),(5,31,24,17),(6,32,21,18),(7,29,22,19),(8,30,23,20)]])
C22.53C24 is a maximal subgroup of
C42.181C23 C42.191C23 C42.201C23 C42.502C23 C42.506C23 C42.511C23 C42.512C23 C42.514C23 C42.516C23 C22.69C25 C22.96C25 C22.99C25 C22.102C25 C22.111C25 C22.113C25 C22.122C25 C22.146C25 C22.155C25
C42.D2p: C42.13D4 C42.114D6 C42.136D6 C42.143D6 C42.166D6 C42.114D10 C42.136D10 C42.143D10 ...
C2p.2+ 1+4: C42.528C23 C42.530C23 C42.74C23 C42.75C23 C42.531C23 C42.533C23 C22.70C25 C22.95C25 ...
C22.53C24 is a maximal quotient of
C23.237C24 C24.212C23 C24.219C23 C24.223C23 C23.345C24 C24.271C23 C23.348C24 C24.276C23 C23.359C24 C23.411C24 C23.413C24 C23.416C24 C23.417C24 C42⋊21D4 C23.457C24 C42.36Q8 C42.37Q8 C24.339C23 C24.346C23 C23.493C24 C24.347C23 C23.496C24 C24.348C23 C23.500C24 C23.502C24 C24.355C23 C24.411C23 C24.412C23 C23.612C24 C23.624C24 C23.651C24 C23.652C24 C23.654C24 C23.671C24 C23.673C24 C23.694C24 C23.696C24 C23.697C24 C23.698C24 C23.703C24 C24.456C23 C23.707C24 C23.708C24
C42.D2p: C42.171D4 C42.178D4 C42.180D4 C42.182D4 C42.114D6 C42.136D6 C42.143D6 C42.166D6 ...
C4⋊C4.D2p: C24.220C23 C24.279C23 C6.672+ 1+4 C10.672+ 1+4 C14.672+ 1+4 ...
Matrix representation of C22.53C24 ►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 |
| 2 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 4 |
| 0 | 0 | 3 | 2 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 3 | 2 |
| 4 | 3 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 4 | 1 |
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],[2,3,0,0,0,3,0,0,0,0,3,3,0,0,4,2],[2,0,0,0,0,2,0,0,0,0,3,3,0,0,0,2],[4,1,0,0,3,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,4,0,0,2,1] >;
C22.53C24 in GAP, Magma, Sage, TeX
C_2^2._{53}C_2^4 % in TeX
G:=Group("C2^2.53C2^4"); // GroupNames label
G:=SmallGroup(64,240);
// by ID
G=gap.SmallGroup(64,240);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,295,650,158,297,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=1,c^2=e^2=b,d^2=b*a=a*b,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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