p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.20D4, C2.D8⋊9C2, C4.Q8⋊12C2, Q8⋊C4⋊8C2, (C2×C4).107D4, C22⋊C8.5C2, C22⋊Q8.7C2, C2.16(C4○D8), C4.34(C4○D4), C4⋊C4.68C22, (C2×C8).40C22, (C2×C4).110C23, C42⋊C2.9C2, C22.106(C2×D4), (C2×Q8).19C22, C2.19(C8.C22), (C22×C4).56C22, C2.16(C22.D4), SmallGroup(64,166)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.20D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >
Character table of C23.20D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 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 | 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 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√2 | √2 | -√-2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 32 points
(2 28)(4 30)(6 32)(8 26)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 22 5 18)(2 12 6 16)(3 20 7 24)(4 10 8 14)(9 31 13 27)(11 29 15 25)(17 32 21 28)(19 30 23 26)
G:=sub<Sym(32)| (2,28)(4,30)(6,32)(8,26)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,5,18)(2,12,6,16)(3,20,7,24)(4,10,8,14)(9,31,13,27)(11,29,15,25)(17,32,21,28)(19,30,23,26)>;
G:=Group( (2,28)(4,30)(6,32)(8,26)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,5,18)(2,12,6,16)(3,20,7,24)(4,10,8,14)(9,31,13,27)(11,29,15,25)(17,32,21,28)(19,30,23,26) );
G=PermutationGroup([[(2,28),(4,30),(6,32),(8,26),(9,18),(10,14),(11,20),(12,16),(13,22),(15,24),(17,21),(19,23)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,22,5,18),(2,12,6,16),(3,20,7,24),(4,10,8,14),(9,31,13,27),(11,29,15,25),(17,32,21,28),(19,30,23,26)]])
C23.20D4 is a maximal subgroup of
C24.115D4 C24.116D4 C24.118D4 (C2×D4).302D4 (C2×D4).303D4 (C2×D4).304D4 C42.229D4 C42.234D4 C24.123D4 C24.124D4 C24.129D4 C24.130D4 C4.2+ 1+4 C4.152+ 1+4 C4.162+ 1+4 C4.172+ 1+4 C42.289D4 C42.292D4
C4⋊C4.D2p: C42.352C23 C42.355C23 C42.357C23 C42.361C23 C42.424C23 C42.425C23 C42.426C23 C42.465C23 ...
C2p.(C4○D8): C42.384D4 C42.451D4 C42.283D4 C42.285D4 C23.15D12 C23.10D20 C23.10D28 ...
C23.20D4 is a maximal quotient of
C24.69D4 C2.D8⋊4C4 C4.Q8⋊9C4 (C2×Q8).109D4 C24.89D4 (C2×C8).171D4 C4⋊C4.Q8
C23.D4p: C23.12D8 C23.15D12 C23.10D20 C23.10D28 ...
C4⋊C4.D2p: C24.71D4 C24.73D4 C4.68(C4×D4) C2.(C4×Q16) C24.85D4 C24.86D4 (C2×C8).24Q8 D6⋊C8.C2 ...
Matrix representation of C23.20D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,0,2,0,0,8,0] >;
C23.20D4 in GAP, Magma, Sage, TeX
C_2^3._{20}D_4 % in TeX
G:=Group("C2^3.20D4"); // GroupNames label
G:=SmallGroup(64,166);
// by ID
G=gap.SmallGroup(64,166);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,199,362,50,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^3>;
// generators/relations
Export
Subgroup lattice of C23.20D4 in TeX
Character table of C23.20D4 in TeX