p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.54C23, C4.642+ 1+4, C8⋊7D4⋊40C2, C8⋊9D4⋊21C2, C8⋊D4⋊40C2, C4⋊C8⋊37C22, C4⋊C4.369D4, C4⋊Q8⋊25C22, D4⋊D4⋊45C2, D8⋊C4⋊24C2, C22⋊D8⋊32C2, D4⋊5D4⋊10C2, D4⋊Q8⋊35C2, (C2×D4).173D4, C2.50(D4○D8), (C4×D4)⋊27C22, C2.D8⋊38C22, C4.Q8⋊27C22, C8⋊C4⋊26C22, D4.27(C4○D4), D4.2D4⋊42C2, C4⋊D4⋊18C22, C4⋊C4.412C23, C22⋊C8⋊33C22, (C2×C8).190C23, (C2×C4).511C24, C22⋊C4.169D4, (C22×C8)⋊33C22, (C2×D8).85C22, C23.328(C2×D4), D4⋊C4⋊55C22, Q8⋊C4⋊45C22, (C2×SD16)⋊33C22, (C2×D4).237C23, (C2×Q8).223C23, C2.147(D4⋊5D4), C42⋊C2⋊25C22, C22⋊Q8.85C22, C23.24D4⋊20C2, C23.37D4⋊15C2, C23.20D4⋊36C2, C23.19D4⋊36C2, C4.4D4.68C22, C22.771(C22×D4), C2.88(D8⋊C22), C22.49C24⋊5C2, (C22×C4).1155C23, (C22×D4).414C22, C42.28C22⋊18C2, (C2×M4(2)).117C22, C4.236(C2×C4○D4), (C2×C4).608(C2×D4), (C2×C4○D4).214C22, SmallGroup(128,2051)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.54C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, ede-1=b2d >
Subgroups: 472 in 207 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, 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.D8, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×D4, C2×C4○D4, C23.24D4, C23.37D4, C8⋊9D4, D8⋊C4, C22⋊D8, D4⋊D4, D4.2D4, C8⋊7D4, C8⋊D4, D4⋊Q8, C23.19D4, C23.20D4, C42.28C22, D4⋊5D4, C22.49C24, C42.54C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○D8, C42.54C23
Character table of C42.54C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 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 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 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 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 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 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 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 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 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 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 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 | -2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 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 | -2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 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 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 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 |
| ρ27 | 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 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 25 18 22)(2 26 19 23)(3 27 20 24)(4 28 17 21)(5 10 29 16)(6 11 30 13)(7 12 31 14)(8 9 32 15)
(1 15 20 11)(2 14 17 10)(3 13 18 9)(4 16 19 12)(5 26 31 21)(6 25 32 24)(7 28 29 23)(8 27 30 22)
(1 20)(2 4)(3 18)(5 12)(6 15)(7 10)(8 13)(9 30)(11 32)(14 29)(16 31)(17 19)(21 26)(22 24)(23 28)(25 27)
(1 22 18 25)(2 21 19 28)(3 24 20 27)(4 23 17 26)(5 10 29 16)(6 9 30 15)(7 12 31 14)(8 11 32 13)
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), (1,25,18,22)(2,26,19,23)(3,27,20,24)(4,28,17,21)(5,10,29,16)(6,11,30,13)(7,12,31,14)(8,9,32,15), (1,15,20,11)(2,14,17,10)(3,13,18,9)(4,16,19,12)(5,26,31,21)(6,25,32,24)(7,28,29,23)(8,27,30,22), (1,20)(2,4)(3,18)(5,12)(6,15)(7,10)(8,13)(9,30)(11,32)(14,29)(16,31)(17,19)(21,26)(22,24)(23,28)(25,27), (1,22,18,25)(2,21,19,28)(3,24,20,27)(4,23,17,26)(5,10,29,16)(6,9,30,15)(7,12,31,14)(8,11,32,13)>;
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), (1,25,18,22)(2,26,19,23)(3,27,20,24)(4,28,17,21)(5,10,29,16)(6,11,30,13)(7,12,31,14)(8,9,32,15), (1,15,20,11)(2,14,17,10)(3,13,18,9)(4,16,19,12)(5,26,31,21)(6,25,32,24)(7,28,29,23)(8,27,30,22), (1,20)(2,4)(3,18)(5,12)(6,15)(7,10)(8,13)(9,30)(11,32)(14,29)(16,31)(17,19)(21,26)(22,24)(23,28)(25,27), (1,22,18,25)(2,21,19,28)(3,24,20,27)(4,23,17,26)(5,10,29,16)(6,9,30,15)(7,12,31,14)(8,11,32,13) );
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)], [(1,25,18,22),(2,26,19,23),(3,27,20,24),(4,28,17,21),(5,10,29,16),(6,11,30,13),(7,12,31,14),(8,9,32,15)], [(1,15,20,11),(2,14,17,10),(3,13,18,9),(4,16,19,12),(5,26,31,21),(6,25,32,24),(7,28,29,23),(8,27,30,22)], [(1,20),(2,4),(3,18),(5,12),(6,15),(7,10),(8,13),(9,30),(11,32),(14,29),(16,31),(17,19),(21,26),(22,24),(23,28),(25,27)], [(1,22,18,25),(2,21,19,28),(3,24,20,27),(4,23,17,26),(5,10,29,16),(6,9,30,15),(7,12,31,14),(8,11,32,13)]])
Matrix representation of C42.54C23 ►in GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [16,2,0,0,0,0,16,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,1,0,0,0,0,15,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[13,8,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,6,0,0,0,0,0,0,0,0,3,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16] >;
C42.54C23 in GAP, Magma, Sage, TeX
C_4^2._{54}C_2^3 % in TeX
G:=Group("C4^2.54C2^3"); // GroupNames label
G:=SmallGroup(128,2051);
// by ID
G=gap.SmallGroup(128,2051);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=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^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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