p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.41C23, C4.512+ 1+4, C8⋊9D4⋊9C2, D4⋊6(C4○D4), D4⋊6D4⋊9C2, D4⋊5D4⋊9C2, C8⋊2D4⋊23C2, C8⋊8D4⋊47C2, C4⋊C8⋊32C22, C4⋊C4.153D4, C4⋊Q8⋊22C22, C22⋊D8⋊30C2, D8⋊C4⋊21C2, D4⋊D4⋊41C2, D4⋊2Q8⋊17C2, (C2×D4).313D4, (C4×D4)⋊25C22, (C2×C8).94C23, C8⋊C4⋊21C22, C4.Q8⋊24C22, D4.2D4⋊40C2, C4⋊D4⋊16C22, C4⋊C4.231C23, C22⋊C8⋊28C22, (C2×C4).498C24, C22⋊C4.163D4, (C2×D8).83C22, C23.473(C2×D4), D4⋊C4⋊54C22, C2.72(D4○SD16), Q8⋊C4⋊42C22, (C2×SD16)⋊52C22, (C2×D4).228C23, C22.8(C8⋊C22), (C2×Q8).213C23, C2.134(D4⋊5D4), C22⋊Q8.77C22, C23.36D4⋊13C2, C23.46D4⋊14C2, C23.47D4⋊14C2, (C2×M4(2))⋊26C22, (C22×C8).361C22, C4.4D4.62C22, C22.758(C22×D4), (C22×C4).1142C23, (C22×D4).409C22, C42.28C22⋊12C2, (C2×C4⋊C4)⋊57C22, C4.223(C2×C4○D4), (C2×C4).595(C2×D4), C2.74(C2×C8⋊C22), (C2×D4⋊C4)⋊42C2, (C2×C4○D4).204C22, SmallGroup(128,2038)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.41C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=eae=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >
Subgroups: 504 in 221 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×D4, C2×C4○D4, C2×C4○D4, C2×D4⋊C4, C23.36D4, C8⋊9D4, D8⋊C4, C22⋊D8, D4⋊D4, D4.2D4, C8⋊8D4, C8⋊2D4, D4⋊2Q8, C23.46D4, C23.47D4, C42.28C22, D4⋊5D4, D4⋊6D4, C42.41C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8⋊C22, D4○SD16, C42.41C23
Character table of C42.41C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | -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 | -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 | -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 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ27 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ28 | 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 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 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 | -2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
►On 32 points
(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 19 23 25)(2 20 24 26)(3 17 21 27)(4 18 22 28)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)
(1 10 3 12)(2 9 4 11)(5 22 7 24)(6 21 8 23)(13 20 15 18)(14 19 16 17)(25 29 27 31)(26 32 28 30)
(1 3)(2 22)(4 24)(5 13)(6 31)(7 15)(8 29)(9 30)(10 14)(11 32)(12 16)(17 25)(18 20)(19 27)(21 23)(26 28)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)
G:=sub<Sym(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), (1,19,23,25)(2,20,24,26)(3,17,21,27)(4,18,22,28)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,10,3,12)(2,9,4,11)(5,22,7,24)(6,21,8,23)(13,20,15,18)(14,19,16,17)(25,29,27,31)(26,32,28,30), (1,3)(2,22)(4,24)(5,13)(6,31)(7,15)(8,29)(9,30)(10,14)(11,32)(12,16)(17,25)(18,20)(19,27)(21,23)(26,28), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)>;
G:=Group( (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), (1,19,23,25)(2,20,24,26)(3,17,21,27)(4,18,22,28)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,10,3,12)(2,9,4,11)(5,22,7,24)(6,21,8,23)(13,20,15,18)(14,19,16,17)(25,29,27,31)(26,32,28,30), (1,3)(2,22)(4,24)(5,13)(6,31)(7,15)(8,29)(9,30)(10,14)(11,32)(12,16)(17,25)(18,20)(19,27)(21,23)(26,28), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31) );
G=PermutationGroup([[(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)], [(1,19,23,25),(2,20,24,26),(3,17,21,27),(4,18,22,28),(5,15,9,32),(6,16,10,29),(7,13,11,30),(8,14,12,31)], [(1,10,3,12),(2,9,4,11),(5,22,7,24),(6,21,8,23),(13,20,15,18),(14,19,16,17),(25,29,27,31),(26,32,28,30)], [(1,3),(2,22),(4,24),(5,13),(6,31),(7,15),(8,29),(9,30),(10,14),(11,32),(12,16),(17,25),(18,20),(19,27),(21,23),(26,28)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31)]])
Matrix representation of C42.41C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,5,12,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.41C23 in GAP, Magma, Sage, TeX
C_4^2._{41}C_2^3 % in TeX
G:=Group("C4^2.41C2^3"); // GroupNames label
G:=SmallGroup(128,2038);
// by ID
G=gap.SmallGroup(128,2038);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
// generators/relations
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