p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4○D4.4D4, (C2×Q8).68D4, (C22×Q8)⋊6C4, C4.59C22≀C2, (C2×D4).267D4, C23.124(C2×D4), D4.14(C22⋊C4), C22.13C22≀C2, Q8.14(C22⋊C4), (C22×C4).26C23, C2.21(C24⋊3C4), C42⋊C22⋊10C2, (C22×Q8).8C22, C23.38D4⋊20C2, C23.32(C22⋊C4), C23.C23⋊3C2, C42⋊C2.1C22, (C2×2- 1+4).2C2, M4(2).8C22⋊9C2, (C2×M4(2)).150C22, (C2×C4○D4)⋊6C4, C4.9(C2×C22⋊C4), (C2×C4).228(C2×D4), (C2×D4).204(C2×C4), (C22×C4).16(C2×C4), (C2×Q8).187(C2×C4), (C2×C4○D4).9C22, (C2×C4).45(C22⋊C4), (C2×C4).177(C22×C4), C22.34(C2×C22⋊C4), SmallGroup(128,528)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C22×Q8)⋊C4
G = < a,b,c,d,e | a2=b2=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, eae-1=bc2, ede-1=bc=cb, bd=db, ebe-1=a, dcd-1=c-1, ece-1=ac2d >
Subgroups: 508 in 268 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C23⋊C4, C4.D4, C4.10D4, Q8⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C22×Q8, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, C23.C23, M4(2).8C22, C23.38D4, C42⋊C22, C2×2- 1+4, (C22×Q8)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, (C22×Q8)⋊C4
Character table of (C22×Q8)⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ27 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 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
(1 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(21 27)(22 28)(23 25)(24 26)
(1 31)(2 32)(3 29)(4 30)(5 27)(6 28)(7 25)(8 26)(9 18)(10 19)(11 20)(12 17)(13 23)(14 24)(15 21)(16 22)
(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 6 3 8)(2 5 4 7)(9 21 11 23)(10 24 12 22)(13 18 15 20)(14 17 16 19)(25 32 27 30)(26 31 28 29)
(2 14)(4 16)(5 23 7 21)(6 30 8 32)(9 28)(10 12)(11 26)(13 15)(17 31 19 29)(18 24 20 22)
G:=sub<Sym(32)| (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,32)(10,29)(11,30)(12,31)(21,27)(22,28)(23,25)(24,26), (1,31)(2,32)(3,29)(4,30)(5,27)(6,28)(7,25)(8,26)(9,18)(10,19)(11,20)(12,17)(13,23)(14,24)(15,21)(16,22), (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), (1,6,3,8)(2,5,4,7)(9,21,11,23)(10,24,12,22)(13,18,15,20)(14,17,16,19)(25,32,27,30)(26,31,28,29), (2,14)(4,16)(5,23,7,21)(6,30,8,32)(9,28)(10,12)(11,26)(13,15)(17,31,19,29)(18,24,20,22)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,15)(6,16)(7,13)(8,14)(9,32)(10,29)(11,30)(12,31)(21,27)(22,28)(23,25)(24,26), (1,31)(2,32)(3,29)(4,30)(5,27)(6,28)(7,25)(8,26)(9,18)(10,19)(11,20)(12,17)(13,23)(14,24)(15,21)(16,22), (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), (1,6,3,8)(2,5,4,7)(9,21,11,23)(10,24,12,22)(13,18,15,20)(14,17,16,19)(25,32,27,30)(26,31,28,29), (2,14)(4,16)(5,23,7,21)(6,30,8,32)(9,28)(10,12)(11,26)(13,15)(17,31,19,29)(18,24,20,22) );
G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,15),(6,16),(7,13),(8,14),(9,32),(10,29),(11,30),(12,31),(21,27),(22,28),(23,25),(24,26)], [(1,31),(2,32),(3,29),(4,30),(5,27),(6,28),(7,25),(8,26),(9,18),(10,19),(11,20),(12,17),(13,23),(14,24),(15,21),(16,22)], [(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)], [(1,6,3,8),(2,5,4,7),(9,21,11,23),(10,24,12,22),(13,18,15,20),(14,17,16,19),(25,32,27,30),(26,31,28,29)], [(2,14),(4,16),(5,23,7,21),(6,30,8,32),(9,28),(10,12),(11,26),(13,15),(17,31,19,29),(18,24,20,22)]])
Matrix representation of (C22×Q8)⋊C4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 16 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 1 | 15 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 1 |
| 1 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 13 | 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 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 | 16 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 16 |
G:=sub<GL(8,GF(17))| [0,0,0,0,0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,1,1,1,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,1,0,0,0,0],[0,0,0,0,1,0,0,16,0,0,0,0,16,0,1,1,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,16,0,0,1,0,0,0,0,1,0,16,16,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,1,0,0,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,4,4,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13],[0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,0,0,0],[1,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16] >;
(C22×Q8)⋊C4 in GAP, Magma, Sage, TeX
(C_2^2\times Q_8)\rtimes C_4
% in TeX
G:=Group("(C2^2xQ8):C4"); // GroupNames label
G:=SmallGroup(128,528);
// by ID
G=gap.SmallGroup(128,528);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,1018,248,2804,1027]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b*c^2,e*d*e^-1=b*c=c*b,b*d=d*b,e*b*e^-1=a,d*c*d^-1=c^-1,e*c*e^-1=a*c^2*d>;
// generators/relations
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