p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.275D4, C42.403C23, C4.1092+ 1+4, C4:D8:26C2, C8:3D4:11C2, C4:C8:19C22, C22:D8:22C2, C4:SD16:10C2, (C2xD8):25C22, (C4xD4):14C22, (C2xC8).65C23, (C4xQ8):14C22, C8:C4:10C22, C22:SD16:11C2, C4:C4.156C23, (C2xC4).415C24, C23.695(C2xD4), (C22xC4).504D4, D4:C4:31C22, C4.126(C8:C22), C42.6C4:13C2, (C2xSD16):23C22, (C2xD4).164C23, C4.4D4:61C22, C22:C8.50C22, (C2xQ8).152C23, C42.C2:38C22, C4:1D4.167C22, C4:D4.193C22, C22.32(C8:C22), (C2xC42).882C22, C22.675(C22xD4), C22:Q8.198C22, C42.29C22:1C2, (C22xC4).1086C23, (C22xD4).391C22, C23.36C23:11C2, C2.86(C22.29C24), (C2xC4:1D4):21C2, (C2xC4).544(C2xD4), C2.58(C2xC8:C22), SmallGroup(128,1949)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.275D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=b2c3 >
Subgroups: 628 in 245 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, D8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C24, C8:C4, C22:C8, D4:C4, C4:C8, C2xC42, C42:C2, C4xD4, C4xD4, C4xQ8, C4:D4, C22:Q8, C22.D4, C4.4D4, C42.C2, C42:2C2, C4:1D4, C4:1D4, C2xD8, C2xSD16, C22xD4, C22xD4, C42.6C4, C22:D8, C22:SD16, C4:D8, C4:SD16, C42.29C22, C8:3D4, C23.36C23, C2xC4:1D4, C42.275D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C8:C22, C22xD4, 2+ 1+4, C22.29C24, C2xC8:C22, C42.275D4
Character table of C42.275D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 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 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
►On 32 points
(1 16 30 19)(2 13 31 24)(3 10 32 21)(4 15 25 18)(5 12 26 23)(6 9 27 20)(7 14 28 17)(8 11 29 22)
(1 7 5 3)(2 29 6 25)(4 31 8 27)(9 18 13 22)(10 16 14 12)(11 20 15 24)(17 23 21 19)(26 32 30 28)
(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 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)
G:=sub<Sym(32)| (1,16,30,19)(2,13,31,24)(3,10,32,21)(4,15,25,18)(5,12,26,23)(6,9,27,20)(7,14,28,17)(8,11,29,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,18,13,22)(10,16,14,12)(11,20,15,24)(17,23,21,19)(26,32,30,28), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)>;
G:=Group( (1,16,30,19)(2,13,31,24)(3,10,32,21)(4,15,25,18)(5,12,26,23)(6,9,27,20)(7,14,28,17)(8,11,29,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,18,13,22)(10,16,14,12)(11,20,15,24)(17,23,21,19)(26,32,30,28), (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), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28) );
G=PermutationGroup([[(1,16,30,19),(2,13,31,24),(3,10,32,21),(4,15,25,18),(5,12,26,23),(6,9,27,20),(7,14,28,17),(8,11,29,22)], [(1,7,5,3),(2,29,6,25),(4,31,8,27),(9,18,13,22),(10,16,14,12),(11,20,15,24),(17,23,21,19),(26,32,30,28)], [(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)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28)]])
Matrix representation of C42.275D4 ►in GL8(F17)
| 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 9 | 1 | 9 |
| 0 | 0 | 0 | 0 | 2 | 0 | 13 | 16 |
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 8 | 16 | 8 |
| 0 | 0 | 0 | 0 | 0 | 1 | 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,15,16,2,0,0,0,0,1,1,9,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,8,1,0,0,0,0,15,0,16,0,0,0,0,0,16,16,8,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,1,0,0] >;
C42.275D4 in GAP, Magma, Sage, TeX
C_4^2._{275}D_4 % in TeX
G:=Group("C4^2.275D4"); // GroupNames label
G:=SmallGroup(128,1949);
// by ID
G=gap.SmallGroup(128,1949);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
// generators/relations
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