p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×Q8).211D4, C42⋊C2.9C4, (C22×C4).28C23, C23.66(C22×C4), (C22×Q8).9C22, M4(2)⋊4C4.2C2, C42⋊C2.6C22, C4.3(C22.D4), C22.8(C42⋊C2), C2.18(C23.34D4), (C2×M4(2)).165C22, C23.32C23.1C2, (C2×C4).236(C2×D4), C22⋊C4.51(C2×C4), (C22×C4).18(C2×C4), (C2×C4).313(C4○D4), (C2×C4).48(C22⋊C4), (C2×C4.10D4).7C2, C22.38(C2×C22⋊C4), SmallGroup(128,562)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×Q8).211D4
G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=ab2c, ab=ba, ac=ca, dad-1=eae-1=ab2, cbc-1=b-1, dbd-1=ebe-1=ab-1, dcd-1=ece-1=b2c, ede-1=b2cd3 >
Subgroups: 204 in 118 conjugacy classes, 50 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, C42⋊C2, C4×Q8, C2×M4(2), C22×Q8, M4(2)⋊4C4, C2×C4.10D4, C23.32C23, (C2×Q8).211D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, (C2×Q8).211D4
Character table of (C2×Q8).211D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -i | i | i | i | i | -i | -i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | i | i | -i | -i | -i | -i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | -i | i | i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | i | -i | i | -i | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | i | -i | -i | -i | -i | i | i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -i | -i | i | i | i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | i | -i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | -i | i | -i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
Generators in S32
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 19 5 23)(2 24 6 20)(3 17 7 21)(4 22 8 18)(9 27 13 31)(10 32 14 28)(11 25 15 29)(12 30 16 26)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)
(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 10 25 22 5 14 29 18)(2 21 26 13 6 17 30 9)(3 16 27 20 7 12 31 24)(4 19 28 11 8 23 32 15)
G:=sub<Sym(32)| (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,19,5,23)(2,24,6,20)(3,17,7,21)(4,22,8,18)(9,27,13,31)(10,32,14,28)(11,25,15,29)(12,30,16,26), (1,29,5,25)(2,26,6,30)(3,31,7,27)(4,28,8,32)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20), (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), (1,10,25,22,5,14,29,18)(2,21,26,13,6,17,30,9)(3,16,27,20,7,12,31,24)(4,19,28,11,8,23,32,15)>;
G:=Group( (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32), (1,19,5,23)(2,24,6,20)(3,17,7,21)(4,22,8,18)(9,27,13,31)(10,32,14,28)(11,25,15,29)(12,30,16,26), (1,29,5,25)(2,26,6,30)(3,31,7,27)(4,28,8,32)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20), (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), (1,10,25,22,5,14,29,18)(2,21,26,13,6,17,30,9)(3,16,27,20,7,12,31,24)(4,19,28,11,8,23,32,15) );
G=PermutationGroup([[(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32)], [(1,19,5,23),(2,24,6,20),(3,17,7,21),(4,22,8,18),(9,27,13,31),(10,32,14,28),(11,25,15,29),(12,30,16,26)], [(1,29,5,25),(2,26,6,30),(3,31,7,27),(4,28,8,32),(9,17,13,21),(10,22,14,18),(11,19,15,23),(12,24,16,20)], [(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)], [(1,10,25,22,5,14,29,18),(2,21,26,13,6,17,30,9),(3,16,27,20,7,12,31,24),(4,19,28,11,8,23,32,15)]])
Matrix representation of (C2×Q8).211D4 ►in GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 | 0 | 16 |
| 7 | 5 | 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 0 | 12 | 10 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 12 | 0 | 5 | 7 |
| 3 | 0 | 0 | 3 | 12 | 5 | 0 | 12 |
| 14 | 3 | 0 | 0 | 5 | 7 | 12 | 0 |
| 3 | 0 | 0 | 3 | 0 | 12 | 12 | 5 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 4 | 0 | 0 | 1 | 16 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 4 | 0 | 0 | 0 | 0 | 1 | 16 |
| 14 | 3 | 0 | 0 | 12 | 0 | 5 | 7 |
| 3 | 14 | 0 | 0 | 12 | 10 | 12 | 0 |
| 3 | 14 | 0 | 0 | 12 | 10 | 5 | 0 |
| 6 | 11 | 0 | 0 | 12 | 10 | 12 | 10 |
| 0 | 5 | 5 | 7 | 0 | 0 | 0 | 0 |
| 12 | 5 | 5 | 12 | 0 | 3 | 14 | 3 |
| 0 | 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 7 | 5 | 0 | 12 | 0 | 3 | 14 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 16 | 1 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 13 | 0 | 0 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,13,0,13,0,1,0,0,0,4,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[7,0,0,5,14,3,14,3,5,5,5,0,3,0,3,0,5,5,12,12,0,0,0,0,7,0,0,10,0,3,0,3,0,0,0,0,12,12,5,0,0,0,0,0,0,5,7,12,0,0,0,0,5,0,12,12,0,0,0,0,7,12,0,5],[0,1,1,1,13,0,13,0,16,0,16,0,4,4,4,4,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[14,3,3,6,0,12,0,7,3,14,14,11,5,5,12,5,0,0,0,0,5,5,12,0,0,0,0,0,7,12,0,12,12,12,12,12,0,0,0,0,0,10,10,10,0,3,0,3,5,12,5,12,0,14,0,14,7,0,0,10,0,3,0,3],[0,13,0,0,0,1,16,0,0,4,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,15,0,16,0,13,0,13,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
(C2×Q8).211D4 in GAP, Magma, Sage, TeX
(C_2\times Q_8)._{211}D_4 % in TeX
G:=Group("(C2xQ8).211D4"); // GroupNames label
G:=SmallGroup(128,562);
// by ID
G=gap.SmallGroup(128,562);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2019,718,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^4=b^2,e^2=a*b^2*c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=e*b*e^-1=a*b^-1,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=b^2*c*d^3>;
// generators/relations
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