p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.406C23, C4.1122+ 1+4, C8⋊4D4⋊10C2, C8⋊3D4⋊13C2, C4⋊D8⋊27C2, C4⋊C8⋊20C22, (C4×C8)⋊10C22, C4⋊C4.129D4, (C2×D8)⋊6C22, C22⋊D8⋊23C2, D4⋊D4⋊28C2, C2.30(D4○D8), (C4×D4)⋊15C22, C4⋊1D4⋊8C22, C22⋊C4.21D4, C8⋊C4⋊11C22, D4.2D4⋊25C2, C4⋊C4.159C23, (C2×C8).161C23, (C2×C4).418C24, Q8⋊C4⋊6C22, C23.290(C2×D4), C42.C2⋊5C22, D4⋊C4⋊32C22, (C2×SD16)⋊24C22, (C2×D4).167C23, C4⋊D4.43C22, C22⋊C8.53C22, (C2×Q8).155C23, C22.29C24⋊16C2, (C22×C4).306C23, C4.4D4.38C22, C22.678(C22×D4), C22.34C24⋊5C2, C42.29C22⋊3C2, C42.7C22⋊10C2, C42.78C22⋊1C2, (C22×D4).392C22, C42⋊C2.157C22, C2.89(C22.29C24), (C2×C4).547(C2×D4), (C2×C4○D4).177C22, SmallGroup(128,1952)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.406C23
G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, cac=dad=a-1, eae=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece=a2c, de=ed >
Subgroups: 540 in 210 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C4.4D4, C42.C2, C4⋊1D4, C2×D8, C2×SD16, C22×D4, C2×C4○D4, C42.7C22, C22⋊D8, D4⋊D4, C4⋊D8, D4.2D4, C42.78C22, C42.29C22, C8⋊4D4, C8⋊3D4, C22.29C24, C22.34C24, C42.406C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D4○D8, C42.406C23
Character table of C42.406C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 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 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 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 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | 4 | -4 | -4 | 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 |
| ρ24 | 4 | 4 | -4 | -4 | 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 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 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 19 25 21)(2 20 26 22)(3 17 27 23)(4 18 28 24)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)
(1 12)(2 11)(3 10)(4 9)(5 28)(6 27)(7 26)(8 25)(13 20)(14 19)(15 18)(16 17)(21 31)(22 30)(23 29)(24 32)
(1 21)(2 24)(3 23)(4 22)(5 7)(9 11)(13 32)(14 31)(15 30)(16 29)(17 27)(18 26)(19 25)(20 28)
(2 26)(4 28)(5 11)(6 8)(7 9)(10 12)(13 32)(14 16)(15 30)(18 24)(20 22)(29 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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,12)(2,11)(3,10)(4,9)(5,28)(6,27)(7,26)(8,25)(13,20)(14,19)(15,18)(16,17)(21,31)(22,30)(23,29)(24,32), (1,21)(2,24)(3,23)(4,22)(5,7)(9,11)(13,32)(14,31)(15,30)(16,29)(17,27)(18,26)(19,25)(20,28), (2,26)(4,28)(5,11)(6,8)(7,9)(10,12)(13,32)(14,16)(15,30)(18,24)(20,22)(29,31)>;
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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,12)(2,11)(3,10)(4,9)(5,28)(6,27)(7,26)(8,25)(13,20)(14,19)(15,18)(16,17)(21,31)(22,30)(23,29)(24,32), (1,21)(2,24)(3,23)(4,22)(5,7)(9,11)(13,32)(14,31)(15,30)(16,29)(17,27)(18,26)(19,25)(20,28), (2,26)(4,28)(5,11)(6,8)(7,9)(10,12)(13,32)(14,16)(15,30)(18,24)(20,22)(29,31) );
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)], [(1,19,25,21),(2,20,26,22),(3,17,27,23),(4,18,28,24),(5,15,9,32),(6,16,10,29),(7,13,11,30),(8,14,12,31)], [(1,12),(2,11),(3,10),(4,9),(5,28),(6,27),(7,26),(8,25),(13,20),(14,19),(15,18),(16,17),(21,31),(22,30),(23,29),(24,32)], [(1,21),(2,24),(3,23),(4,22),(5,7),(9,11),(13,32),(14,31),(15,30),(16,29),(17,27),(18,26),(19,25),(20,28)], [(2,26),(4,28),(5,11),(6,8),(7,9),(10,12),(13,32),(14,16),(15,30),(18,24),(20,22),(29,31)]])
Matrix representation of C42.406C23 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 3 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,0,14,0,0,0,0,0,0,11,0,0,0,0,0,0,14,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3],[1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C42.406C23 in GAP, Magma, Sage, TeX
C_4^2._{406}C_2^3 % in TeX
G:=Group("C4^2.406C2^3"); // GroupNames label
G:=SmallGroup(128,1952);
// by ID
G=gap.SmallGroup(128,1952);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,c*a*c=d*a*d=a^-1,e*a*e=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations
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