p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.47D4, C22.6SD16, C4.Q8⋊11C2, (C2×C4).39D4, C22⋊C8.6C2, C22⋊Q8.5C2, Q8⋊C4⋊13C2, C4.32(C4○D4), C4⋊C4.66C22, (C2×C8).39C22, C2.13(C2×SD16), (C2×C4).108C23, C22.104(C2×D4), (C2×Q8).17C22, C2.18(C8.C22), (C22×C4).54C22, C2.14(C22.D4), (C2×C4⋊C4).16C2, SmallGroup(64,164)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.47D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >
Character table of C23.47D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 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 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| 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 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(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 25 13 29)(11 31 15 27)(17 26 21 30)(19 32 23 28)
G:=sub<Sym(32)| (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(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)(9,25,13,29)(11,31,15,27)(17,26,21,30)(19,32,23,28)>;
G:=Group( (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(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)(9,25,13,29)(11,31,15,27)(17,26,21,30)(19,32,23,28) );
G=PermutationGroup([[(2,30),(4,32),(6,26),(8,28),(9,22),(11,24),(13,18),(15,20)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(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),(9,25,13,29),(11,31,15,27),(17,26,21,30),(19,32,23,28)]])
C23.47D4 is a maximal subgroup of
C24.115D4 C24.183D4 C24.118D4 (C2×D4).302D4 (C2×D4).304D4 C42.226D4 C42.231D4 C42.232D4 C23⋊4SD16 C24.123D4 C24.127D4 C24.128D4 C4.162+ 1+4 C4.192+ 1+4 C42.284D4 C42.288D4 C42.290D4
C4⋊C4.D2p: C24.14D4 C4⋊C4.12D4 (C2×C4).SD16 C24.15D4 C42.354C23 C42.359C23 C42.424C23 C42.426C23 ...
C2p.(C2×SD16): C42.222D4 C42.281D4 C23.39D12 C23.34D20 C23.34D28 ...
C23.47D4 is a maximal quotient of
C4.Q8⋊10C4 (C2×C4).19Q16 C24.89D4 (C2×C8).170D4 (C2×C4).28D8
C23.D4p: C23.36D8 C23.39D12 C23.34D20 C23.34D28 ...
C4⋊C4.D2p: C24.159D4 C24.160D4 C4.68(C4×D4) C24.85D4 C2.(C8⋊3Q8) D6.1SD16 D6.2SD16 C4⋊C4.231D6 ...
Matrix representation of C23.47D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 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 |
| 4 | 8 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 16 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 10 |
| 0 | 0 | 10 | 1 |
G:=sub<GL(4,GF(17))| [1,16,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[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],[4,13,0,0,8,13,0,0,0,0,12,12,0,0,5,12],[16,0,0,0,15,1,0,0,0,0,16,10,0,0,10,1] >;
C23.47D4 in GAP, Magma, Sage, TeX
C_2^3._{47}D_4 % in TeX
G:=Group("C2^3.47D4"); // GroupNames label
G:=SmallGroup(64,164);
// by ID
G=gap.SmallGroup(64,164);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,50,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,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 C23.47D4 in TeX
Character table of C23.47D4 in TeX