p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.4D8, C4.3SD16, C42.76C22, (C4xC8):5C2, C4:Q8:5C2, C2.9(C2xD8), (C2xC4).74D4, D4:C4:3C2, C4:1D4.4C2, C4.13(C4oD4), C4:C4.16C22, (C2xC8).66C22, C2.14(C2xSD16), (C2xC4).111C23, C2.9(C4.4D4), (C2xD4).24C22, C22.107(C2xD4), SmallGroup(64,167)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.4D8
G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 129 in 59 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C42, C4:C4, C4:C4, C2xC8, C2xD4, C2xD4, C2xQ8, C4xC8, D4:C4, C4:1D4, C4:Q8, C4.4D8
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2xD4, C4oD4, C4.4D4, C2xD8, C2xSD16, C4.4D8
Character table of C4.4D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -√2 | -√2 | -√2 | √2 | √2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | √2 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | -2i | 0 | 2i | 0 | complex lifted from C4oD4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 0 | 2i | 0 | 2i | complex lifted from C4oD4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 2i | 0 | -2i | 0 | complex lifted from C4oD4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | 0 | -2i | 0 | -2i | complex lifted from C4oD4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
►On 32 points
(1 21 27 10)(2 22 28 11)(3 23 29 12)(4 24 30 13)(5 17 31 14)(6 18 32 15)(7 19 25 16)(8 20 26 9)
(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 26 27 8)(2 7 28 25)(3 32 29 6)(4 5 30 31)(9 21 20 10)(11 19 22 16)(12 15 23 18)(13 17 24 14)
G:=sub<Sym(32)| (1,21,27,10)(2,22,28,11)(3,23,29,12)(4,24,30,13)(5,17,31,14)(6,18,32,15)(7,19,25,16)(8,20,26,9), (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), (1,26,27,8)(2,7,28,25)(3,32,29,6)(4,5,30,31)(9,21,20,10)(11,19,22,16)(12,15,23,18)(13,17,24,14)>;
G:=Group( (1,21,27,10)(2,22,28,11)(3,23,29,12)(4,24,30,13)(5,17,31,14)(6,18,32,15)(7,19,25,16)(8,20,26,9), (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), (1,26,27,8)(2,7,28,25)(3,32,29,6)(4,5,30,31)(9,21,20,10)(11,19,22,16)(12,15,23,18)(13,17,24,14) );
G=PermutationGroup([[(1,21,27,10),(2,22,28,11),(3,23,29,12),(4,24,30,13),(5,17,31,14),(6,18,32,15),(7,19,25,16),(8,20,26,9)], [(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)], [(1,26,27,8),(2,7,28,25),(3,32,29,6),(4,5,30,31),(9,21,20,10),(11,19,22,16),(12,15,23,18),(13,17,24,14)]])
C4.4D8 is a maximal subgroup of
D4:3D8 Q8:6SD16 Q8:3D8 C42.189C23 D4.D8 C42.201C23 Q8.D8 Q8:3SD16 D4.2SD16 Q8.2SD16 C42.248C23 C42.249C23 C82:3C2 C42.355D4 C42.366C23 C42.240D4 C42.243D4 C42.263D4 C42.278D4 C42.279D4 C42.280D4 D4:7SD16 C42.469C23 C42.473C23 C42.482C23 Q8:4D8 C42.502C23 Q8:8SD16 C42.506C23 C42.507C23 C42.509C23 C42.514C23 C42.516C23
C4p.D8: C82:5C2 C8.2D8 C4.5D24 C12.16D8 C12.D8 C4.5D40 C20.16D8 C20.D8 ...
C4p:Q8:C2: C42.391C23 C42.423C23 Dic3.SD16 Dic5.5D8 Dic7.SD16 ...
C8:pD4:C2: C8:5D8 C8:6SD16 C42.664C23 C8:3D8 C42.365D4 C42.366D4 C42.259D4 C42.261D4 ...
C2.(C8:pD4): D4.2D8 Q8.2D8 C42.252C23 C42.253C23 C82:12C2 C8:5SD16 C42.666C23 C42.667C23 ...
C4.4D8 is a maximal quotient of
C42.55Q8 C2.(C8:7D4) C42.432D4 C42.436D4 C4:C4:7D4 (C2xC4).24D8 (C2xC4).28D8
C4.D8p: C4.4D16 C4.5D24 C4.5D40 C4.5D56 ...
C4p.SD16: C4.SD32 C8.22SD16 C8.12SD16 C8.13SD16 C8.14SD16 C12.16D8 C12.D8 C20.16D8 ...
C4:C4.D2p: (C2xD4):Q8 Dic3.SD16 Dic5.5D8 Dic7.SD16 ...
Matrix representation of C4.4D8 ►in GL4(F17) generated by
| 16 | 2 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 13 | 0 |
| 7 | 10 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 7 | 10 | 0 | 0 |
| 12 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,0,13,0,0,13,0],[7,12,0,0,10,0,0,0,0,0,0,1,0,0,1,0],[7,12,0,0,10,10,0,0,0,0,0,16,0,0,1,0] >;
C4.4D8 in GAP, Magma, Sage, TeX
C_4._4D_8
% in TeX
G:=Group("C4.4D8"); // GroupNames label
G:=SmallGroup(64,167);
// by ID
G=gap.SmallGroup(64,167);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,50,963,117,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations
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