p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4oD16, C4oQ32, C4oSD32, D16:3C2, Q32:3C2, C8.13D4, C4.20D8, SD32:3C2, C8.9C23, C22.1D8, C16.6C22, D8.2C22, Q16.2C22, (C2xC16):6C2, C4oD8:1C2, C2.15(C2xD8), (C2xC4).84D4, C4.10(C2xD4), (C2xC8).89C22, SmallGroup(64,189)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4oD16
G = < a,b,c | a4=c2=1, b8=a2, ab=ba, ac=ca, cbc=a2b7 >
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C2xD8, C4oD16
Character table of C4oD16
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | √2 | -√-2 | -√2 | √-2 | ζ167+ζ16 | ζ165+ζ163 | -ζ167+ζ16 | ζ1613+ζ1611 | -ζ165+ζ163 | ζ1615+ζ169 | ζ165-ζ163 | ζ167-ζ16 | complex faithful |
| ρ16 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -√2 | √-2 | √2 | -√-2 | ζ165+ζ163 | ζ1615+ζ169 | ζ165-ζ163 | ζ167+ζ16 | -ζ167+ζ16 | ζ1613+ζ1611 | ζ167-ζ16 | -ζ165+ζ163 | complex faithful |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | √2 | √-2 | -√2 | -√-2 | ζ167+ζ16 | ζ165+ζ163 | ζ167-ζ16 | ζ1613+ζ1611 | ζ165-ζ163 | ζ1615+ζ169 | -ζ165+ζ163 | -ζ167+ζ16 | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -√2 | -√-2 | √2 | √-2 | ζ165+ζ163 | ζ1615+ζ169 | -ζ165+ζ163 | ζ167+ζ16 | ζ167-ζ16 | ζ1613+ζ1611 | -ζ167+ζ16 | ζ165-ζ163 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | √2 | -√-2 | -√2 | √-2 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ167-ζ16 | ζ165+ζ163 | ζ165-ζ163 | ζ167+ζ16 | -ζ165+ζ163 | -ζ167+ζ16 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -√2 | √-2 | √2 | -√-2 | ζ1613+ζ1611 | ζ167+ζ16 | -ζ165+ζ163 | ζ1615+ζ169 | ζ167-ζ16 | ζ165+ζ163 | -ζ167+ζ16 | ζ165-ζ163 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -√2 | -√-2 | √2 | √-2 | ζ1613+ζ1611 | ζ167+ζ16 | ζ165-ζ163 | ζ1615+ζ169 | -ζ167+ζ16 | ζ165+ζ163 | ζ167-ζ16 | -ζ165+ζ163 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | √2 | √-2 | -√2 | -√-2 | ζ1615+ζ169 | ζ1613+ζ1611 | -ζ167+ζ16 | ζ165+ζ163 | -ζ165+ζ163 | ζ167+ζ16 | ζ165-ζ163 | ζ167-ζ16 | complex faithful |
►On 32 points
(1 27 9 19)(2 28 10 20)(3 29 11 21)(4 30 12 22)(5 31 13 23)(6 32 14 24)(7 17 15 25)(8 18 16 26)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)
G:=sub<Sym(32)| (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)>;
G:=Group( (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,20)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27) );
G=PermutationGroup([[(1,27,9,19),(2,28,10,20),(3,29,11,21),(4,30,12,22),(5,31,13,23),(6,32,14,24),(7,17,15,25),(8,18,16,26)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,20),(18,19),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27)]])
C4oD16 is a maximal subgroup of
D16.C4 D16:3C4 D16:C22
D16p:C2: C4oD32 C32:C22 D48:7C2 D48:5C2 D80:7C2 D80:5C2 D112:7C2 Q32:3D7 ...
D2p.D8: D4oD16 D4oSD32 Q8oD16 D16:3S3 D6.2D8 D16:3D5 SD32:3D5 D16:3D7 ...
C4p.D8: C8oD16 D16:5C4 Q64:C2 Q16.D6 C40.30C23 C56.30C23 ...
C4oD16 is a maximal quotient of
C23.24D8 C23.25D8 C4xD16 C4xSD32 C4xQ32 D8:8D4 Q16.5D4 C16:7D4 C16:8D4 D8.Q8 Q16.Q8 C23.19D8 C23.20D8 C8.22SD16 C16.5Q8
D8.D2p: D8.10D4 D8.5D4 D16:3S3 D6.2D8 Q16.D6 D16:3D5 SD32:3D5 C40.30C23 ...
C8.D4p: C8.21D8 D48:7C2 D80:7C2 D112:7C2 ...
C16.D2p: C16.19D4 D48:5C2 D80:5C2 Q32:3D7 ...
Matrix representation of C4oD16 ►in GL2(F17) generated by
| 13 | 0 |
| 0 | 13 |
| 13 | 11 |
| 6 | 13 |
| 13 | 11 |
| 11 | 4 |
G:=sub<GL(2,GF(17))| [13,0,0,13],[13,6,11,13],[13,11,11,4] >;
C4oD16 in GAP, Magma, Sage, TeX
C_4\circ D_{16} % in TeX
G:=Group("C4oD16"); // GroupNames label
G:=SmallGroup(64,189);
// by ID
G=gap.SmallGroup(64,189);
# by ID
G:=PCGroup([6,-2,2,2,-2,-2,-2,121,158,579,297,165,1444,730,88]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^8=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^7>;
// generators/relations
Export
Subgroup lattice of C4oD16 in TeX
Character table of C4oD16 in TeX