p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.408C23, C4.1142+ 1+4, C8⋊5D4⋊25C2, C4⋊C8⋊21C22, (C4×C8)⋊43C22, C4⋊C4.131D4, C4⋊Q8⋊12C22, C4⋊SD16⋊11C2, C8.2D4⋊13C2, C22⋊C4.23D4, (C4×Q8)⋊15C22, C8⋊C4⋊12C22, D4.7D4⋊29C2, C22⋊SD16⋊12C2, C4⋊C4.161C23, (C2×C8).330C23, (C2×C4).420C24, Q8.D4⋊25C2, (C2×Q16)⋊26C22, C23.292(C2×D4), C42.C2⋊6C22, D4⋊C4⋊33C22, C2.46(D4○SD16), Q8⋊C4⋊48C22, (C2×SD16)⋊44C22, (C2×D4).169C23, C4⋊1D4.68C22, C22⋊C8.55C22, (C2×Q8).157C23, C22⋊Q8.43C22, (C22×C4).308C23, C4.4D4.40C22, C22.680(C22×D4), C42.7C22⋊12C2, C22.35C24⋊5C2, C22.29C24.15C2, C42.29C22⋊4C2, (C22×D4).393C22, C42.78C22⋊15C2, C42⋊C2.159C22, C2.91(C22.29C24), (C2×C4).549(C2×D4), (C2×C4○D4).179C22, SmallGroup(128,1954)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.408C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b2, ab=ba, cac-1=dad=a-1, eae=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, de=ed >
Subgroups: 460 in 198 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, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C42.7C22, C22⋊SD16, D4.7D4, C4⋊SD16, Q8.D4, C42.78C22, C42.29C22, C8⋊5D4, C8.2D4, C22.29C24, C22.35C24, C42.408C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D4○SD16, C42.408C23
Character table of C42.408C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 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 | 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 | -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 | -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 | 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 | -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 | 4 | -4 | 4 | -4 | 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 |
| ρ22 | 4 | -4 | 4 | -4 | 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 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | 0 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 0 | 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 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 8 25 12)(2 7 26 11)(3 6 27 10)(4 5 28 9)(13 20 30 22)(14 19 31 21)(15 18 32 24)(16 17 29 23)
(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), (1,8,25,12)(2,7,26,11)(3,6,27,10)(4,5,28,9)(13,20,30,22)(14,19,31,21)(15,18,32,24)(16,17,29,23), (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), (1,8,25,12)(2,7,26,11)(3,6,27,10)(4,5,28,9)(13,20,30,22)(14,19,31,21)(15,18,32,24)(16,17,29,23), (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)], [(1,8,25,12),(2,7,26,11),(3,6,27,10),(4,5,28,9),(13,20,30,22),(14,19,31,21),(15,18,32,24),(16,17,29,23)], [(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 C42.408C23 ►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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 15 | 16 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 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 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 1 |
| 0 | 0 | 0 | 0 | 2 | 1 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 | 0 | 12 |
| 0 | 0 | 0 | 0 | 7 | 7 | 5 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 16 | 1 | 1 |
| 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 | 16 | 16 |
| 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))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,15,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,16,0,16,0,0,0,0,1,16,1,0],[0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,5,5,5,0,0,0,0,12,5,5,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,0,16,0,0,16] >;
C42.408C23 in GAP, Magma, Sage, TeX
C_4^2._{408}C_2^3 % in TeX
G:=Group("C4^2.408C2^3"); // GroupNames label
G:=SmallGroup(128,1954);
// by ID
G=gap.SmallGroup(128,1954);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e=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,d*e=e*d>;
// generators/relations
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