p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.9C23, C42.90C22, C22.26C24, (C2×C4)⋊8D4, C4○(C4⋊Q8), (C4×D4)⋊9C2, C4⋊Q8⋊19C2, C4⋊1(C4○D4), C4○(C4⋊1D4), C4.82(C2×D4), C4○3(C4⋊D4), C4⋊1D4⋊10C2, C4⋊D4⋊20C2, (C2×C42)⋊12C2, C22.1(C2×D4), C4○2(C4.4D4), C4.4D4⋊18C2, C4⋊C4.72C22, C2.12(C22×D4), (C2×C4).160C23, (C2×D4).63C22, (C2×Q8).56C22, C22⋊C4.14C22, (C22×C4).125C22, (C2×C4○D4)⋊3C2, (C2×C4)○(C4⋊D4), (C2×C4)○(C4⋊1D4), C2.13(C2×C4○D4), SmallGroup(64,213)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.26C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=b, f2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, df=fd, ef=fe >
Subgroups: 233 in 155 conjugacy classes, 85 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C22.26C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.26C24
Character table of C22.26C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | -2 | 0 | 2i | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | -2 | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | 0 | -2i | -2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | 0 | 2i | -2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 22)(10 23)(11 24)(12 21)(25 31)(26 32)(27 29)(28 30)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 22)(2 21)(3 24)(4 23)(5 32)(6 31)(7 30)(8 29)(9 15)(10 14)(11 13)(12 16)(17 27)(18 26)(19 25)(20 28)
(1 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 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 19 15 6)(2 20 16 7)(3 17 13 8)(4 18 14 5)(9 31 22 25)(10 32 23 26)(11 29 24 27)(12 30 21 28)
G:=sub<Sym(32)| (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(25,31)(26,32)(27,29)(28,30), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,22)(2,21)(3,24)(4,23)(5,32)(6,31)(7,30)(8,29)(9,15)(10,14)(11,13)(12,16)(17,27)(18,26)(19,25)(20,28), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (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), (1,19,15,6)(2,20,16,7)(3,17,13,8)(4,18,14,5)(9,31,22,25)(10,32,23,26)(11,29,24,27)(12,30,21,28)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(25,31)(26,32)(27,29)(28,30), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,22)(2,21)(3,24)(4,23)(5,32)(6,31)(7,30)(8,29)(9,15)(10,14)(11,13)(12,16)(17,27)(18,26)(19,25)(20,28), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (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), (1,19,15,6)(2,20,16,7)(3,17,13,8)(4,18,14,5)(9,31,22,25)(10,32,23,26)(11,29,24,27)(12,30,21,28) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,18),(6,19),(7,20),(8,17),(9,22),(10,23),(11,24),(12,21),(25,31),(26,32),(27,29),(28,30)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,22),(2,21),(3,24),(4,23),(5,32),(6,31),(7,30),(8,29),(9,15),(10,14),(11,13),(12,16),(17,27),(18,26),(19,25),(20,28)], [(1,25),(2,26),(3,27),(4,28),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)], [(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)], [(1,19,15,6),(2,20,16,7),(3,17,13,8),(4,18,14,5),(9,31,22,25),(10,32,23,26),(11,29,24,27),(12,30,21,28)]])
C22.26C24 is a maximal subgroup of
C43⋊C2 C42⋊8D4 M4(2)⋊23D4 C42.694C23 C42.300C23 C42.301C23 C42.313C23 C22.33C25 C22.38C25 C22.44C25 C22.49C25 D4×C4○D4 C22.69C25 C22.70C25 C22.72C25 C22.83C25 C4⋊2+ 1+4 C4⋊2- 1+4 C22.87C25 C22.88C25 C22.89C25 C22.95C25 C22.96C25 C22.97C25 C22.100C25 C22.101C25 C22.103C25 C22.106C25 C22.108C25 C22.111C25 C22.118C25 C22.135C25 C22.136C25 C22.137C25 C22.138C25 C22.139C25 C22.140C25 C22.143C25 C22.147C25 C22.148C25 C22.149C25 C22.151C25
C42.D2p: C42.45D4 C42.400D4 C42.315D4 C42.53D4 C42.403D4 C42.55D4 C42.59D4 C42.61D4 ...
(C2×C4p)⋊D4: C42.681C23 C42.266C23 (C2×C12)⋊17D4 (C2×C20)⋊17D4 (C2×C28)⋊17D4 ...
(C2×D4).D2p: M4(2)⋊19D4 C4⋊Q8⋊29C4 (C2×D4).135D4 C12⋊(C4○D4) C20⋊(C4○D4) C28⋊(C4○D4) ...
C22.26C24 is a maximal quotient of
C23.167C24 C43⋊9C2 C23.179C24 C4×C4⋊D4 C4×C4⋊1D4 C23.288C24 C42⋊15D4 C23.295C24 C23.328C24 C24.262C23 C24.263C23 C23.352C24 C24.282C23 C23.364C24 C23.391C24 C23.396C24 C23.398C24 C24.308C23 C23.400C24 C23.406C24 C23.412C24 C23.419C24 C24.311C23 C23.443C24 C24.326C23 C24.327C23 C23.455C24 C23.456C24 C23.457C24 C23.458C24 C24.331C23 C24.332C23 C24.583C23 C42⋊23D4 C23.502C24 C42⋊24D4 C42⋊26D4 C24.377C23 C42⋊32D4 C24.378C23 C42⋊46D4 C24.598C23 C42⋊47D4 C43⋊12C2 C43⋊13C2 C42.385C23 C42.386C23 C42.387C23 C42.388C23 C42.389C23
C42.D2p: C4×C4.4D4 C4×C4⋊Q8 C42.162D4 C42.168D4 C42.174D4 C42.175D4 C42.184D4 C42.185D4 ...
C4p⋊Q8⋊C2: C42.390C23 C42.391C23 C12⋊(C4○D4) C20⋊(C4○D4) C28⋊(C4○D4) ...
(C2×D4).D2p: C23.322C24 C24.264C23 C23.345C24 C42⋊21D4 (C2×C12)⋊17D4 (C2×C20)⋊17D4 (C2×C28)⋊17D4 ...
Matrix representation of C22.26C24 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 4 | 0 | 0 |
| 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,3,0,0,4,2,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,3,4,0,0,0,0,0,3,0,0,2,0],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;
C22.26C24 in GAP, Magma, Sage, TeX
C_2^2._{26}C_2^4 % in TeX
G:=Group("C2^2.26C2^4"); // GroupNames label
G:=SmallGroup(64,213);
// by ID
G=gap.SmallGroup(64,213);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,650,158,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=b,f^2=a,a*b=b*a,d*c*d=a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations
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