p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.50C22, C23.20C23, C22.49C24, C2.182+ 1+4, C4⋊Q8⋊16C2, (C4×D4)⋊21C2, C4⋊D4⋊16C2, C4.38(C4○D4), C4.4D4⋊13C2, C4⋊C4.83C22, (C2×C4).32C23, C42⋊C2⋊17C2, (C2×D4).36C22, (C2×Q8).33C22, C22⋊C4.24C22, (C22×C4).74C22, C2.28(C2×C4○D4), SmallGroup(64,236)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.49C24
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=b, e2=f2=a, ab=ba, 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, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C22.49C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C22.49C24
Character table of C22.49C24
| 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 | 2 | 0 | 0 | 0 | 2i | 2i | 0 | -2i | 0 | 0 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2i | 0 | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2i | -2i | 0 | 0 | -2 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | -2i | 0 | 0 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 2 | 0 | -2i | 2i | 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 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | -2i | 0 | 2i | 0 | 0 | 2i | -2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2i | 2i | 0 | 0 | -2 | 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 | orthogonal lifted from 2+ 1+4 |
►On 32 points
(1 15)(2 16)(3 13)(4 14)(5 20)(6 17)(7 18)(8 19)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)
(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 31)(2 27)(3 29)(4 25)(5 23)(6 9)(7 21)(8 11)(10 18)(12 20)(13 28)(14 30)(15 26)(16 32)(17 24)(19 22)
(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 6 15 17)(2 7 16 18)(3 8 13 19)(4 5 14 20)(9 28 24 29)(10 25 21 30)(11 26 22 31)(12 27 23 32)
(1 17 15 6)(2 7 16 18)(3 19 13 8)(4 5 14 20)(9 31 24 26)(10 27 21 32)(11 29 22 28)(12 25 23 30)
G:=sub<Sym(32)| (1,15)(2,16)(3,13)(4,14)(5,20)(6,17)(7,18)(8,19)(9,24)(10,21)(11,22)(12,23)(25,30)(26,31)(27,32)(28,29), (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), (1,31)(2,27)(3,29)(4,25)(5,23)(6,9)(7,21)(8,11)(10,18)(12,20)(13,28)(14,30)(15,26)(16,32)(17,24)(19,22), (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), (1,6,15,17)(2,7,16,18)(3,8,13,19)(4,5,14,20)(9,28,24,29)(10,25,21,30)(11,26,22,31)(12,27,23,32), (1,17,15,6)(2,7,16,18)(3,19,13,8)(4,5,14,20)(9,31,24,26)(10,27,21,32)(11,29,22,28)(12,25,23,30)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,20)(6,17)(7,18)(8,19)(9,24)(10,21)(11,22)(12,23)(25,30)(26,31)(27,32)(28,29), (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), (1,31)(2,27)(3,29)(4,25)(5,23)(6,9)(7,21)(8,11)(10,18)(12,20)(13,28)(14,30)(15,26)(16,32)(17,24)(19,22), (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), (1,6,15,17)(2,7,16,18)(3,8,13,19)(4,5,14,20)(9,28,24,29)(10,25,21,30)(11,26,22,31)(12,27,23,32), (1,17,15,6)(2,7,16,18)(3,19,13,8)(4,5,14,20)(9,31,24,26)(10,27,21,32)(11,29,22,28)(12,25,23,30) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,20),(6,17),(7,18),(8,19),(9,24),(10,21),(11,22),(12,23),(25,30),(26,31),(27,32),(28,29)], [(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)], [(1,31),(2,27),(3,29),(4,25),(5,23),(6,9),(7,21),(8,11),(10,18),(12,20),(13,28),(14,30),(15,26),(16,32),(17,24),(19,22)], [(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)], [(1,6,15,17),(2,7,16,18),(3,8,13,19),(4,5,14,20),(9,28,24,29),(10,25,21,30),(11,26,22,31),(12,27,23,32)], [(1,17,15,6),(2,7,16,18),(3,19,13,8),(4,5,14,20),(9,31,24,26),(10,27,21,32),(11,29,22,28),(12,25,23,30)]])
C22.49C24 is a maximal subgroup of
C42.185C23 C42.195C23 C42.207C23 C42.489C23 C42.490C23 C42.61C23 C42.62C23 C42.495C23 C42.498C23 C22.64C25 C22.82C25 C22.83C25 C22.107C25 C22.113C25 C22.133C25 C22.134C25 C22.154C25 C22.155C25
C42.D2p: C42.17D4 C42.97D6 C42.117D6 C42.138D6 C42.179D6 C42.97D10 C42.117D10 C42.138D10 ...
C2p.2+ 1+4: C42.468C23 C42.470C23 C42.54C23 C42.56C23 C42.474C23 C42.478C23 C22.72C25 C22.95C25 ...
C22.49C24 is a maximal quotient of
C24.208C23 C23.236C24 C24.217C23 C23.251C24 C24.221C23 C24.249C23 C24.254C23 C23.359C24 C23.374C24 C24.293C23 C23.385C24 C23.395C24 C23.397C24 C23.400C24 C23.408C24 C23.413C24 C23.418C24 C23.426C24 C23.429C24 C23.431C24 C23.432C24 C42⋊18D4 C23.455C24 C42.37Q8 C23.472C24 C24.338C23 C23.493C24 C42⋊25D4 C42⋊26D4 C42⋊9Q8 C24.393C23 C24.406C23 C23.631C24 C23.633C24 C23.637C24 C24.430C23 C23.656C24 C23.669C24 C23.675C24 C23.681C24 C23.683C24 C24.450C23 C23.685C24 C23.695C24 C23.703C24 C23.708C24 C23.709C24
C42.D2p: C42.173D4 C42.97D6 C42.117D6 C42.138D6 C42.179D6 C42.97D10 C42.117D10 C42.138D10 ...
C4⋊C4.D2p: C24.267C23 C23.391C24 C6.452+ 1+4 C10.452+ 1+4 C14.452+ 1+4 ...
Matrix representation of C22.49C24 ►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 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
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],[4,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,4,0,0,4,0,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,4],[3,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1] >;
C22.49C24 in GAP, Magma, Sage, TeX
C_2^2._{49}C_2^4 % in TeX
G:=Group("C2^2.49C2^4"); // GroupNames label
G:=SmallGroup(64,236);
// by ID
G=gap.SmallGroup(64,236);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,86,297,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=b,e^2=f^2=a,a*b=b*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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