p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.7Q8, C23.22D4, C23.56C23, C24.24C22, (C22×C4)⋊8C4, C22⋊1(C4⋊C4), C4⋊2(C22⋊C4), (C2×C4).115D4, (C23×C4).6C2, C22.9(C2×Q8), C2.1(C4⋊D4), C23.25(C2×C4), C22.30(C2×D4), C2.1(C22⋊Q8), C2.C42⋊1C2, C2.5(C42⋊C2), C22.15(C4○D4), C22.29(C22×C4), (C22×C4).87C22, (C2×C4⋊C4)⋊1C2, C2.4(C2×C4⋊C4), (C2×C4).69(C2×C4), C2.5(C2×C22⋊C4), (C2×C22⋊C4).3C2, SmallGroup(64,61)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.7Q8
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=bd2, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 185 in 117 conjugacy classes, 57 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C23.7Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8
Character table of C23.7Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | -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 | 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 | 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 | -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 | -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 | 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 | 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 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | 2 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -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 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | -2 | 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 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(1 27)(2 28)(3 25)(4 26)(5 11)(6 12)(7 9)(8 10)(13 32)(14 29)(15 30)(16 31)(17 21)(18 22)(19 23)(20 24)
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(17 27)(18 28)(19 25)(20 26)
(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 8 25 12)(2 7 26 11)(3 6 27 10)(4 5 28 9)(13 20 30 22)(14 19 31 21)(15 18 32 24)(16 17 29 23)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,32)(6,29)(7,30)(8,31)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,27)(2,28)(3,25)(4,26)(5,11)(6,12)(7,9)(8,10)(13,32)(14,29)(15,30)(16,31)(17,21)(18,22)(19,23)(20,24), (1,21)(2,22)(3,23)(4,24)(5,15)(6,16)(7,13)(8,14)(9,32)(10,29)(11,30)(12,31)(17,27)(18,28)(19,25)(20,26), (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), (1,8,25,12)(2,7,26,11)(3,6,27,10)(4,5,28,9)(13,20,30,22)(14,19,31,21)(15,18,32,24)(16,17,29,23)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,32)(6,29)(7,30)(8,31)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,27)(2,28)(3,25)(4,26)(5,11)(6,12)(7,9)(8,10)(13,32)(14,29)(15,30)(16,31)(17,21)(18,22)(19,23)(20,24), (1,21)(2,22)(3,23)(4,24)(5,15)(6,16)(7,13)(8,14)(9,32)(10,29)(11,30)(12,31)(17,27)(18,28)(19,25)(20,26), (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), (1,8,25,12)(2,7,26,11)(3,6,27,10)(4,5,28,9)(13,20,30,22)(14,19,31,21)(15,18,32,24)(16,17,29,23) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,32),(6,29),(7,30),(8,31),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,27),(2,28),(3,25),(4,26),(5,11),(6,12),(7,9),(8,10),(13,32),(14,29),(15,30),(16,31),(17,21),(18,22),(19,23),(20,24)], [(1,21),(2,22),(3,23),(4,24),(5,15),(6,16),(7,13),(8,14),(9,32),(10,29),(11,30),(12,31),(17,27),(18,28),(19,25),(20,26)], [(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)], [(1,8,25,12),(2,7,26,11),(3,6,27,10),(4,5,28,9),(13,20,30,22),(14,19,31,21),(15,18,32,24),(16,17,29,23)]])
C23.7Q8 is a maximal subgroup of
C23.C42 C24.167C23 C24.175C23 C24.176C23 C25.85C22 C23.165C24 C23.167C24 C4×C4⋊D4 C4×C22⋊Q8 C24.542C23 C23.194C24 C23.195C24 C24.192C23 C24.545C23 C24.547C23 C23.201C24 C42.159D4 C42⋊13D4 C24.198C23 C23.211C24 C24.204C23 D4×C22⋊C4 C24.549C23 Q8×C22⋊C4 C23.223C24 C23.224C24 C23.226C24 C23.227C24 C24.208C23 C23.229C24 D4×C4⋊C4 C23.234C24 C23.235C24 C23.236C24 C24.225C23 C23.259C24 C24.227C23 C24.244C23 C23.308C24 C23.309C24 C24.249C23 C23.315C24 C23.316C24 C24.252C23 C23.318C24 C24.563C23 C24.254C23 C23.321C24 C23.322C24 C23.323C24 C24.258C23 C23.327C24 C24⋊4Q8 C24.567C23 C24.267C23 C24.568C23 C24.268C23 C24.569C23 C23.350C24 C23.354C24 C24.278C23 C23.360C24 C23.367C24 C23.368C24 C24.572C23 C23.380C24 C24.573C23 C23.385C24 C24.299C23 C24.300C23 C24.301C23 C23.390C24 C23.391C24 C23.392C24 C24.577C23 C24.304C23 C23.395C24 C23.396C24 C23.397C24 C23.398C24 C23.405C24 C23.410C24 C24.311C23 C23.422C24 C23.430C24 C23.431C24 C23.434C24 C42⋊17D4 C42.165D4 C42⋊18D4 C42.166D4 C42.170D4 C23.449C24 C24.326C23 C42.172D4 C42.173D4 C24.583C23 C24.584C23 C24.338C23 C24.341C23 C23.478C24 C23.479C24 C24.360C23 C24.361C23 C24⋊10D4 C24.587C23 C23.524C24 C23.525C24 C24⋊5Q8 C23.527C24 C42.187D4 C42.188D4 C23.530C24 C23.546C24 C23.559C24 C24.379C23 C23.567C24 C23.571C24 C23.572C24 C24.393C23 C24.394C23 C23.590C24 C23.591C24 C23.592C24 C23.593C24 C24.401C23 C23.595C24 C24.403C23 C23.602C24 C23.603C24 C24.408C23 C23.605C24 C23.606C24 C23.607C24 C23.608C24 C23.615C24 C23.617C24 C23.622C24 C23.637C24 C24.426C23 C24.427C23 C23.640C24 C23.641C24 C24.428C23 C23.643C24 C24.430C23 C23.645C24 C24.432C23 C23.647C24 C24.434C23 C24.435C23 C23.656C24 C23.668C24 C24.445C23 C23.678C24 C23.679C24 C24.448C23 C23.681C24 C24.450C23 C23.686C24 C23.687C24 C23.688C24 C24.459C23 C23.714C24 C23.741C24 C24⋊13D4 C24⋊8Q8 C42.439D4 C24.598C23 C24.599C23 C42.440D4
C24.D2p: C23.8D8 C23.30D8 C24.58D4 C24.60D4 C24.61D4 C23.35D8 C24.155D4 C24.65D4 ...
D2p⋊(C4⋊C4): C23.231C24 D6⋊(C4⋊C4) C4⋊(D6⋊C4) D10⋊2(C4⋊C4) D10⋊4(C4⋊C4) D10⋊6(C4⋊C4) D14⋊(C4⋊C4) C4⋊(D14⋊C4) ...
C23.7Q8 is a maximal quotient of
C24.624C23 C24.625C23 C24.631C23 C42.425D4 C42.95D4 C24.167C23 C42.96D4 C42.97D4 C42.99D4 C42.100D4 C42.101D4 C24.19Q8 C24.9Q8 (C2×D4).24Q8 (C2×C8).103D4 C8○D4⋊C4 C4○D4.4Q8 C4○D4.5Q8
C24.D2p: C24.17Q8 C24.5Q8 C24.133D4 C23.22D8 C24.67D4 C24.55D6 C24.75D6 C24.44D10 ...
D2p⋊(C4⋊C4): C42.98D4 C42.102D4 D6⋊(C4⋊C4) C4⋊(D6⋊C4) D10⋊2(C4⋊C4) D10⋊4(C4⋊C4) D10⋊6(C4⋊C4) D14⋊(C4⋊C4) ...
Matrix representation of C23.7Q8 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 3 | 0 | 0 |
| 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,1,0,0,0,3,4,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,4,4,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3,4] >;
C23.7Q8 in GAP, Magma, Sage, TeX
C_2^3._7Q_8
% in TeX
G:=Group("C2^3.7Q8"); // GroupNames label
G:=SmallGroup(64,61);
// by ID
G=gap.SmallGroup(64,61);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,55,362]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=b*d^2,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*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=d^-1>;
// generators/relations
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