p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.472C23, C4.682+ 1+4, C8⋊D4⋊43C2, C8⋊6D4⋊11C2, C4⋊C8⋊39C22, (C4×C8)⋊48C22, C4⋊C4.162D4, D4.Q8⋊39C2, (C4×SD16)⋊55C2, D4⋊5D4.5C2, (C2×D4).176D4, C8.D4⋊27C2, (C4×Q8)⋊29C22, C4.Q8⋊29C22, C2.D8⋊40C22, D4.29(C4○D4), C22⋊SD16⋊23C2, D4.7D4⋊48C2, C4⋊C4.415C23, C22⋊C8⋊35C22, (C2×C4).515C24, (C2×C8).103C23, Q8.D4⋊44C2, C22⋊C4.172D4, (C2×Q16)⋊33C22, C23.332(C2×D4), C22⋊Q8⋊20C22, C2.80(D4○SD16), Q8⋊C4⋊50C22, (C4×D4).165C22, (C2×D4).425C23, C4⋊D4.89C22, (C2×Q8).226C23, C2.151(D4⋊5D4), C42.C2⋊12C22, C23.20D4⋊40C2, C23.36D4⋊22C2, C23.37D4⋊17C2, C23.46D4⋊18C2, (C2×M4(2))⋊31C22, (C22×C4).328C23, C4.4D4.71C22, C22.775(C22×D4), D4⋊C4.121C22, C2.91(D8⋊C22), C22.46C24⋊6C2, (C2×SD16).159C22, (C22×D4).416C22, C42.78C22⋊21C2, C42⋊C2.195C22, (C2×C4⋊C4)⋊62C22, C4.240(C2×C4○D4), (C2×C4).928(C2×D4), (C2×C4○D4).217C22, SmallGroup(128,2055)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.472C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=a-1b2, dad=ab2, eae=a-1, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >
Subgroups: 432 in 201 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, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C2×M4(2), C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, C8⋊6D4, C4×SD16, C22⋊SD16, D4.7D4, Q8.D4, C8⋊D4, C8.D4, D4.Q8, C23.46D4, C23.20D4, C42.78C22, D4⋊5D4, C22.46C24, C42.472C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○SD16, C42.472C23
Character table of C42.472C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | -2 | 0 | 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 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | -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 | -4 | 0 | 0 | 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 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊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 | 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 | 2√-2 | 0 | -2√-2 | 0 | 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 18 25 24)(2 19 26 21)(3 20 27 22)(4 17 28 23)(5 15 29 12)(6 16 30 9)(7 13 31 10)(8 14 32 11)
(1 30 27 8)(2 5 28 31)(3 32 25 6)(4 7 26 29)(9 20 14 24)(10 21 15 17)(11 18 16 22)(12 23 13 19)
(1 27)(2 4)(3 25)(5 10)(6 14)(7 12)(8 16)(9 32)(11 30)(13 29)(15 31)(17 21)(18 20)(19 23)(22 24)(26 28)
(1 4)(2 3)(5 30)(6 29)(7 32)(8 31)(9 15)(10 14)(11 13)(12 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)
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), (1,18,25,24)(2,19,26,21)(3,20,27,22)(4,17,28,23)(5,15,29,12)(6,16,30,9)(7,13,31,10)(8,14,32,11), (1,30,27,8)(2,5,28,31)(3,32,25,6)(4,7,26,29)(9,20,14,24)(10,21,15,17)(11,18,16,22)(12,23,13,19), (1,27)(2,4)(3,25)(5,10)(6,14)(7,12)(8,16)(9,32)(11,30)(13,29)(15,31)(17,21)(18,20)(19,23)(22,24)(26,28), (1,4)(2,3)(5,30)(6,29)(7,32)(8,31)(9,15)(10,14)(11,13)(12,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)>;
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), (1,18,25,24)(2,19,26,21)(3,20,27,22)(4,17,28,23)(5,15,29,12)(6,16,30,9)(7,13,31,10)(8,14,32,11), (1,30,27,8)(2,5,28,31)(3,32,25,6)(4,7,26,29)(9,20,14,24)(10,21,15,17)(11,18,16,22)(12,23,13,19), (1,27)(2,4)(3,25)(5,10)(6,14)(7,12)(8,16)(9,32)(11,30)(13,29)(15,31)(17,21)(18,20)(19,23)(22,24)(26,28), (1,4)(2,3)(5,30)(6,29)(7,32)(8,31)(9,15)(10,14)(11,13)(12,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27) );
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)], [(1,18,25,24),(2,19,26,21),(3,20,27,22),(4,17,28,23),(5,15,29,12),(6,16,30,9),(7,13,31,10),(8,14,32,11)], [(1,30,27,8),(2,5,28,31),(3,32,25,6),(4,7,26,29),(9,20,14,24),(10,21,15,17),(11,18,16,22),(12,23,13,19)], [(1,27),(2,4),(3,25),(5,10),(6,14),(7,12),(8,16),(9,32),(11,30),(13,29),(15,31),(17,21),(18,20),(19,23),(22,24),(26,28)], [(1,4),(2,3),(5,30),(6,29),(7,32),(8,31),(9,15),(10,14),(11,13),(12,16),(17,18),(19,20),(21,22),(23,24),(25,28),(26,27)]])
Matrix representation of C42.472C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 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 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 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 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 4 | 13 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,9,13,0,0,0,0,0,0,13,13,0,0,0,0,8,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,9,13,0,0,0,0,0,0,4,4,0,0,0,0,9,13] >;
C42.472C23 in GAP, Magma, Sage, TeX
C_4^2._{472}C_2^3 % in TeX
G:=Group("C4^2.472C2^3"); // GroupNames label
G:=SmallGroup(128,2055);
// by ID
G=gap.SmallGroup(128,2055);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,e*a*e=a^-1,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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