p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊3C4, C42.11C22, C8⋊3(C2×C4), (C4×D4)⋊2C2, D4⋊2(C2×C4), C8⋊C4⋊2C2, C4.Q8⋊3C2, (C2×D8).6C2, C2.17(C4×D4), C4.6(C4○D4), (C2×C4).103D4, D4⋊C4⋊16C2, C4⋊C4.55C22, C2.5(C8⋊C22), (C2×C4).78C23, (C2×C8).15C22, C4.14(C22×C4), C22.56(C2×D4), (C2×D4).53C22, 2-Sylow(PSigmaL(2,81)), Aut(SD32), SmallGroup(64,123)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊C4
G = < a,b,c | a8=b2=c4=1, bab=a-1, cac-1=a5, cbc-1=a4b >
Subgroups: 125 in 66 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, D8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22, D8⋊C4
Character table of D8⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | 1 | -i | -i | i | i | i | -1 | -i | 1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | i | 1 | i | i | -i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | -i | i | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | i | -i | i | 1 | i | -i | -i | i | -i | 1 | i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | i | -i | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | i | -i | i | -i | 1 | -i | i | i | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | 1 | i | i | -i | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | -i | 1 | -i | -i | i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2i | -2i | 2i | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2i | 2i | -2i | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
►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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
(1 22 25 13)(2 19 26 10)(3 24 27 15)(4 21 28 12)(5 18 29 9)(6 23 30 14)(7 20 31 11)(8 17 32 16)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19), (1,22,25,13)(2,19,26,10)(3,24,27,15)(4,21,28,12)(5,18,29,9)(6,23,30,14)(7,20,31,11)(8,17,32,16)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19), (1,22,25,13)(2,19,26,10)(3,24,27,15)(4,21,28,12)(5,18,29,9)(6,23,30,14)(7,20,31,11)(8,17,32,16) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,32),(7,31),(8,30),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)], [(1,22,25,13),(2,19,26,10),(3,24,27,15),(4,21,28,12),(5,18,29,9),(6,23,30,14),(7,20,31,11),(8,17,32,16)]])
D8⋊C4 is a maximal subgroup of
C42.352C23 C42.353C23 C42.356C23 C42.359C23 C42.385C23 C42.388C23 C42.391C23 D8⋊9D4 D8⋊10D4 C42.41C23 C42.42C23 C42.53C23 C42.54C23 C42.59C23 C42.61C23 C42.507C23 C42.511C23 C42.514C23 C42.517C23 C42.74C23 C42.531C23
D8p⋊C4: D16⋊C4 D24⋊C4 D24⋊9C4 D40⋊9C4 D40⋊15C4 D40⋊C4 D56⋊C4 D56⋊9C4 ...
C42.D2p: C42.383D4 C4×C8⋊C22 C42.227D4 C42.229D4 C42.232D4 C42.233D4 C42.257D4 C42.260D4 ...
D4p⋊(C2×C4): C42.275C23 C42.277C23 C42.281C23 D4⋊S3⋊C4 D4⋊D5⋊6C4 D4⋊D7⋊C4 ...
(C2p×D8).C2: D8⋊4Q8 D8⋊5Q8 D8⋊Dic3 D8⋊Dic5 D8⋊Dic7 ...
D8⋊C4 is a maximal quotient of
D8⋊5C8 D4⋊2M4(2) C8⋊3M4(2) C8⋊C42 C2.D8⋊5C4 C4.(C4×Q8)
D8p⋊C4: D16⋊C4 D24⋊C4 D24⋊9C4 D40⋊9C4 D40⋊15C4 D40⋊C4 D56⋊C4 D56⋊9C4 ...
C42.D2p: D4⋊C42 Q32⋊C4 C42.48D6 C42.48D10 C42.48D14 ...
(Cp×D8)⋊C4: (C2×C4)⋊9D8 (C2×D8)⋊10C4 D8⋊Dic3 D8⋊Dic5 D8⋊Dic7 ...
C2p.(C4×D4): D4⋊(C4⋊C4) C4.67(C4×D4) C2.(C8⋊2D4) D4⋊S3⋊C4 D4⋊D5⋊6C4 D4⋊D7⋊C4 ...
Matrix representation of D8⋊C4 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 8 | 9 |
| 0 | 0 | 5 | 7 | 4 | 0 |
| 0 | 0 | 8 | 9 | 0 | 10 |
| 0 | 0 | 4 | 0 | 12 | 10 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,5,8,4,0,0,7,7,9,0,0,0,8,4,0,12,0,0,9,0,10,10],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
D8⋊C4 in GAP, Magma, Sage, TeX
D_8\rtimes C_4
% in TeX
G:=Group("D8:C4"); // GroupNames label
G:=SmallGroup(64,123);
// by ID
G=gap.SmallGroup(64,123);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,230,963,489,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations
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