p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.474C23, C4.702+ 1+4, D42⋊10C2, (C4×D8)⋊43C2, C4⋊D8⋊39C2, C8⋊2D4⋊28C2, C8⋊6D4⋊13C2, C4⋊C8⋊41C22, (C4×C8)⋊38C22, C4⋊C4.372D4, C4⋊Q8⋊27C22, C22⋊D8⋊34C2, D4⋊2Q8⋊19C2, (C2×D4).322D4, C2.52(D4○D8), C4.4D8⋊21C2, (C2×D8)⋊32C22, (C4×D4)⋊29C22, C22⋊C4.55D4, C2.D8⋊61C22, C4.Q8⋊30C22, D4.31(C4○D4), D4⋊C4⋊7C22, C4⋊C4.417C23, C4⋊D4⋊20C22, C4.47(C8⋊C22), C22⋊C8⋊37C22, (C2×C8).191C23, (C2×C4).517C24, C23.334(C2×D4), (C2×D4).241C23, C4⋊1D4.91C22, C2.153(D4⋊5D4), C42⋊C2⋊27C22, C23.37D4⋊19C2, C23.19D4⋊39C2, (C2×M4(2))⋊33C22, (C22×C4).330C23, C22.777(C22×D4), C22.49C24⋊7C2, (C22×D4).418C22, C4.242(C2×C4○D4), (C2×C4).612(C2×D4), C2.79(C2×C8⋊C22), SmallGroup(128,2057)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.474C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=a-1, dad=ab2, eae-1=a-1b2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, ede-1=b2d >
Subgroups: 576 in 230 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×D8, C22×D4, C22×D4, C23.37D4, C8⋊6D4, C4×D8, C22⋊D8, C4⋊D8, C8⋊2D4, D4⋊2Q8, C23.19D4, C4.4D8, D42, C22.49C24, C42.474C23
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○D8, C42.474C23
Character table of C42.474C23
| 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 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 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 | 0 | -4 | 4 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
►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 28 21 18)(2 25 22 19)(3 26 23 20)(4 27 24 17)(5 12 15 30)(6 9 16 31)(7 10 13 32)(8 11 14 29)
(1 10 3 12)(2 9 4 11)(5 28 7 26)(6 27 8 25)(13 20 15 18)(14 19 16 17)(21 32 23 30)(22 31 24 29)
(1 3)(2 24)(4 22)(5 10)(6 29)(7 12)(8 31)(9 14)(11 16)(13 30)(15 32)(17 19)(18 26)(20 28)(21 23)(25 27)
(1 24 21 4)(2 3 22 23)(5 8 15 14)(6 13 16 7)(9 32 31 10)(11 30 29 12)(17 18 27 28)(19 20 25 26)
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), (1,28,21,18)(2,25,22,19)(3,26,23,20)(4,27,24,17)(5,12,15,30)(6,9,16,31)(7,10,13,32)(8,11,14,29), (1,10,3,12)(2,9,4,11)(5,28,7,26)(6,27,8,25)(13,20,15,18)(14,19,16,17)(21,32,23,30)(22,31,24,29), (1,3)(2,24)(4,22)(5,10)(6,29)(7,12)(8,31)(9,14)(11,16)(13,30)(15,32)(17,19)(18,26)(20,28)(21,23)(25,27), (1,24,21,4)(2,3,22,23)(5,8,15,14)(6,13,16,7)(9,32,31,10)(11,30,29,12)(17,18,27,28)(19,20,25,26)>;
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), (1,28,21,18)(2,25,22,19)(3,26,23,20)(4,27,24,17)(5,12,15,30)(6,9,16,31)(7,10,13,32)(8,11,14,29), (1,10,3,12)(2,9,4,11)(5,28,7,26)(6,27,8,25)(13,20,15,18)(14,19,16,17)(21,32,23,30)(22,31,24,29), (1,3)(2,24)(4,22)(5,10)(6,29)(7,12)(8,31)(9,14)(11,16)(13,30)(15,32)(17,19)(18,26)(20,28)(21,23)(25,27), (1,24,21,4)(2,3,22,23)(5,8,15,14)(6,13,16,7)(9,32,31,10)(11,30,29,12)(17,18,27,28)(19,20,25,26) );
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)], [(1,28,21,18),(2,25,22,19),(3,26,23,20),(4,27,24,17),(5,12,15,30),(6,9,16,31),(7,10,13,32),(8,11,14,29)], [(1,10,3,12),(2,9,4,11),(5,28,7,26),(6,27,8,25),(13,20,15,18),(14,19,16,17),(21,32,23,30),(22,31,24,29)], [(1,3),(2,24),(4,22),(5,10),(6,29),(7,12),(8,31),(9,14),(11,16),(13,30),(15,32),(17,19),(18,26),(20,28),(21,23),(25,27)], [(1,24,21,4),(2,3,22,23),(5,8,15,14),(6,13,16,7),(9,32,31,10),(11,30,29,12),(17,18,27,28),(19,20,25,26)]])
Matrix representation of C42.474C23 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1] >;
C42.474C23 in GAP, Magma, Sage, TeX
C_4^2._{474}C_2^3 % in TeX
G:=Group("C4^2.474C2^3"); // GroupNames label
G:=SmallGroup(128,2057);
// by ID
G=gap.SmallGroup(128,2057);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,e*a*e^-1=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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