p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.6D8, D4⋊3SD16, C42.200C23, D42.2C2, D4⋊C8⋊20C2, (C4×C8)⋊3C22, C4⋊C4.54D4, C4.28(C2×D8), C4⋊Q8⋊2C22, C4⋊C8⋊43C22, D4⋊Q8⋊1C2, C4.4D8⋊4C2, C4.D8⋊8C2, (C2×D4).254D4, C4.31(C2×SD16), D4.D4⋊30C2, C4.61(C8⋊C22), (C4×D4).31C22, C2.14(C22⋊D8), C4⋊1D4.17C22, C2.14(C22⋊SD16), C22.166C22≀C2, C2.12(D4.9D4), (C2×C4).957(C2×D4), SmallGroup(128,371)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.D8
G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 464 in 154 conjugacy classes, 38 normal (32 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×4], C22, C22 [×22], C8 [×3], C2×C4 [×3], C2×C4 [×7], D4 [×4], D4 [×16], Q8 [×2], C23 [×14], C42, C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×3], SD16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×14], C2×Q8, C24 [×2], C4×C8, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4⋊1D4, C4⋊Q8, C2×SD16, C22×D4 [×2], D4⋊C8 [×2], C4.D8, D4.D4, D4⋊Q8, C4.4D8, D42, D4.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], SD16 [×2], C2×D4 [×3], C22≀C2, C2×D8, C2×SD16, C8⋊C22 [×2], C22⋊D8, C22⋊SD16, D4.9D4, D4.D8
Character table of D4.D8
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 0 | 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 |
ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | 2 | 2 | 2 | 0 | 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 |
ρ15 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
ρ16 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
ρ17 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
ρ18 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
ρ19 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
ρ20 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
ρ21 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
ρ22 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
ρ23 | 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 | orthogonal lifted from C8⋊C22 |
ρ24 | 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 C8⋊C22 |
ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | complex lifted from D4.9D4 |
(1 32 19 13)(2 25 20 14)(3 26 21 15)(4 27 22 16)(5 28 23 9)(6 29 24 10)(7 30 17 11)(8 31 18 12)
(1 9)(2 6)(3 30)(4 18)(5 13)(7 26)(8 22)(10 25)(11 21)(12 16)(14 29)(15 17)(19 28)(20 24)(23 32)(27 31)
(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 14 19 25)(2 32 20 13)(3 12 21 31)(4 30 22 11)(5 10 23 29)(6 28 24 9)(7 16 17 27)(8 26 18 15)
G:=sub<Sym(32)| (1,32,19,13)(2,25,20,14)(3,26,21,15)(4,27,22,16)(5,28,23,9)(6,29,24,10)(7,30,17,11)(8,31,18,12), (1,9)(2,6)(3,30)(4,18)(5,13)(7,26)(8,22)(10,25)(11,21)(12,16)(14,29)(15,17)(19,28)(20,24)(23,32)(27,31), (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), (1,14,19,25)(2,32,20,13)(3,12,21,31)(4,30,22,11)(5,10,23,29)(6,28,24,9)(7,16,17,27)(8,26,18,15)>;
G:=Group( (1,32,19,13)(2,25,20,14)(3,26,21,15)(4,27,22,16)(5,28,23,9)(6,29,24,10)(7,30,17,11)(8,31,18,12), (1,9)(2,6)(3,30)(4,18)(5,13)(7,26)(8,22)(10,25)(11,21)(12,16)(14,29)(15,17)(19,28)(20,24)(23,32)(27,31), (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), (1,14,19,25)(2,32,20,13)(3,12,21,31)(4,30,22,11)(5,10,23,29)(6,28,24,9)(7,16,17,27)(8,26,18,15) );
G=PermutationGroup([(1,32,19,13),(2,25,20,14),(3,26,21,15),(4,27,22,16),(5,28,23,9),(6,29,24,10),(7,30,17,11),(8,31,18,12)], [(1,9),(2,6),(3,30),(4,18),(5,13),(7,26),(8,22),(10,25),(11,21),(12,16),(14,29),(15,17),(19,28),(20,24),(23,32),(27,31)], [(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)], [(1,14,19,25),(2,32,20,13),(3,12,21,31),(4,30,22,11),(5,10,23,29),(6,28,24,9),(7,16,17,27),(8,26,18,15)])
Matrix representation of D4.D8 ►in GL4(𝔽17) generated by
0 | 1 | 0 | 0 |
16 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | 16 | 0 |
0 | 0 | 0 | 16 |
5 | 12 | 0 | 0 |
5 | 5 | 0 | 0 |
0 | 0 | 3 | 3 |
0 | 0 | 14 | 3 |
5 | 12 | 0 | 0 |
12 | 12 | 0 | 0 |
0 | 0 | 3 | 3 |
0 | 0 | 3 | 14 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,12,5,0,0,0,0,3,14,0,0,3,3],[5,12,0,0,12,12,0,0,0,0,3,3,0,0,3,14] >;
D4.D8 in GAP, Magma, Sage, TeX
D_4.D_8
% in TeX
G:=Group("D4.D8");
// GroupNames label
G:=SmallGroup(128,371);
// by ID
G=gap.SmallGroup(128,371);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
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