p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.48D4, C22.3Q16, C2.D8⋊8C2, (C2×C4).40D4, C2.8(C2×Q16), Q8⋊C4⋊7C2, C22⋊C8.4C2, (C2×C8).8C22, C22⋊Q8.6C2, C4.33(C4○D4), C4⋊C4.67C22, C2.19(C8⋊C22), (C2×C4).109C23, C22.105(C2×D4), (C2×Q8).18C22, (C22×C4).55C22, C2.15(C22.D4), (C2×C4⋊C4).17C2, SmallGroup(64,165)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.48D4
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=bcd3 >
Character table of C23.48D4
| 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 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
►On 32 points
(2 31)(4 25)(6 27)(8 29)(10 20)(12 22)(14 24)(16 18)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 18 5 22)(2 15 6 11)(3 24 7 20)(4 13 8 9)(10 32 14 28)(12 30 16 26)(17 27 21 31)(19 25 23 29)
G:=sub<Sym(32)| (2,31)(4,25)(6,27)(8,29)(10,20)(12,22)(14,24)(16,18), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,18,5,22)(2,15,6,11)(3,24,7,20)(4,13,8,9)(10,32,14,28)(12,30,16,26)(17,27,21,31)(19,25,23,29)>;
G:=Group( (2,31)(4,25)(6,27)(8,29)(10,20)(12,22)(14,24)(16,18), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,18,5,22)(2,15,6,11)(3,24,7,20)(4,13,8,9)(10,32,14,28)(12,30,16,26)(17,27,21,31)(19,25,23,29) );
G=PermutationGroup([[(2,31),(4,25),(6,27),(8,29),(10,20),(12,22),(14,24),(16,18)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)], [(1,18,5,22),(2,15,6,11),(3,24,7,20),(4,13,8,9),(10,32,14,28),(12,30,16,26),(17,27,21,31),(19,25,23,29)]])
C23.48D4 is a maximal subgroup of
C24.115D4 C24.183D4 C24.118D4 (C2×D4).302D4 (C2×D4).303D4 C42.224D4 C42.225D4 C42.228D4 C42.235D4 C24.121D4 C23⋊3Q16 C24.126D4 C24.129D4 C4.162+ 1+4 C4.182+ 1+4 C42.284D4 C42.288D4 C42.291D4
C4⋊C4.D2p: C24.17D4 C4⋊C4.18D4 C4⋊C4.20D4 C24.18D4 C42.354C23 C42.358C23 C42.424C23 C42.425C23 ...
(C2×C2p).Q16: C42.282D4 C23.40D12 C23.35D20 C23.35D28 ...
C23.48D4 is a maximal quotient of
C24.157D4 C24.88D4
(C2×C2p).Q16: C2.D8⋊5C4 C2.(C4×Q16) (C2×C4).19Q16 (C2×C8).1Q8 (C2×C8).60D4 (C2×C4).23Q16 C23.40D12 C4⋊C4.230D6 ...
C4⋊C4.D2p: C23.37D8 C24.160D4 C24.86D4 D6.Q16 D6.2Q16 D10.7Q16 D10.8Q16 D14.Q16 ...
Matrix representation of C23.48D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 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 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 12 | 7 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[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],[0,13,0,0,4,0,0,0,0,0,11,3,0,0,11,0],[0,1,0,0,1,0,0,0,0,0,10,12,0,0,10,7] >;
C23.48D4 in GAP, Magma, Sage, TeX
C_2^3._{48}D_4 % in TeX
G:=Group("C2^3.48D4"); // GroupNames label
G:=SmallGroup(64,165);
// by ID
G=gap.SmallGroup(64,165);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,194,1444,376,88]);
// 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*c*d^3>;
// generators/relations
Export
Subgroup lattice of C23.48D4 in TeX
Character table of C23.48D4 in TeX