p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8:10D4, C42.34C23, C4.1212+ 1+4, C2.51D42, C8:9D4:4C2, C8.30(C2xD4), D4:5D4:2C2, D4:6D4:2C2, C8:8D4:18C2, C4:C8:26C22, C4:C4.355D4, D4.19(C2xD4), C4:Q8:15C22, D8:C4:20C2, D4:D4:33C2, (C2xD4).160D4, C8.D4:19C2, C8.2D4:17C2, (C4xD4):18C22, (C2xC8).84C23, C22:C4.39D4, C4.81(C22xD4), C23.99(C2xD4), C4.Q8:23C22, C8:C4:17C22, D4.7D4:33C2, D4.D4:18C2, C22:SD16:17C2, D4.2D4:36C2, C4:C4.206C23, C22:C8:22C22, (C2xC4).465C24, (C22xC8):24C22, (C2xQ16):28C22, C22:Q8:10C22, D4:C4:36C22, C2.58(D4oSD16), Q8:C4:38C22, (C2xSD16):80C22, (C2xD4).205C23, (C2xD8).165C22, C4:D4.57C22, (C2xQ8).192C23, C4.4D4.50C22, C22.725(C22xD4), C2.75(D8:C22), (C22xC4).1117C23, (C22xD4).397C22, (C2xM4(2)).100C22, (C2xC4oD8):23C2, (C2xC8:C22):28C2, (C2xC4).589(C2xD4), (C2xC4oD4):13C22, SmallGroup(128,1999)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8:10D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, cac-1=dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >
Subgroups: 544 in 250 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C8:C4, C22:C8, D4:C4, Q8:C4, C4:C8, C4.Q8, C2xC22:C4, C2xC4:C4, C4xD4, C22wrC2, C4:D4, C4:D4, C22:Q8, C22.D4, C4.4D4, C4:Q8, C22xC8, C2xM4(2), C2xD8, C2xSD16, C2xQ16, C4oD8, C8:C22, C22xD4, C2xC4oD4, C8:9D4, D8:C4, D4:D4, C22:SD16, D4.7D4, D4.D4, D4.2D4, C8:8D4, C8.D4, C8.2D4, D4:5D4, D4:6D4, C2xC4oD8, C2xC8:C22, D8:10D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, 2+ 1+4, D42, D8:C22, D4oSD16, D8:10D4
Character table of D8:10D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 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 | -2 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -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 | -4i | 0 | 0 | 4i | 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 | 0 | 0 | 4i | 0 | 0 | -4i | 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 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4oSD16 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 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 | complex lifted from D4oSD16 |
►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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)
(1 17 29 13)(2 20 30 16)(3 23 31 11)(4 18 32 14)(5 21 25 9)(6 24 26 12)(7 19 27 15)(8 22 28 10)
(2 4)(3 7)(6 8)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)(26 28)(27 31)(30 32)
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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21), (1,17,29,13)(2,20,30,16)(3,23,31,11)(4,18,32,14)(5,21,25,9)(6,24,26,12)(7,19,27,15)(8,22,28,10), (2,4)(3,7)(6,8)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18)(26,28)(27,31)(30,32)>;
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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21), (1,17,29,13)(2,20,30,16)(3,23,31,11)(4,18,32,14)(5,21,25,9)(6,24,26,12)(7,19,27,15)(8,22,28,10), (2,4)(3,7)(6,8)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18)(26,28)(27,31)(30,32) );
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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21)], [(1,17,29,13),(2,20,30,16),(3,23,31,11),(4,18,32,14),(5,21,25,9),(6,24,26,12),(7,19,27,15),(8,22,28,10)], [(2,4),(3,7),(6,8),(9,21),(10,24),(11,19),(12,22),(13,17),(14,20),(15,23),(16,18),(26,28),(27,31),(30,32)]])
Matrix representation of D8:10D4 ►in GL6(F17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 5 | 7 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 12 | 10 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,5,0,0,0,0,7,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,0,5,0,0,0,0,10,0,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,1,0,0,0,0,0,15,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;
D8:10D4 in GAP, Magma, Sage, TeX
D_8\rtimes_{10}D_4 % in TeX
G:=Group("D8:10D4"); // GroupNames label
G:=SmallGroup(128,1999);
// by ID
G=gap.SmallGroup(128,1999);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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