p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.64C25, C23.127C24, C24.498C23, C42.594C23, C4○(D4⋊3Q8), C4⋊Q8⋊87C22, C4○2(D4⋊5D4), D4⋊10(C4○D4), C4○2(D4⋊6D4), D4⋊6D4⋊47C2, D4⋊5D4⋊43C2, D4⋊3Q8⋊47C2, (C4×D4)⋊31C22, (C4×Q8)⋊34C22, D4○2(C42⋊C2), C4⋊D4⋊76C22, C4⋊C4.473C23, (C2×C4).166C24, (C2×C42)⋊54C22, C22⋊Q8⋊91C22, (C2×D4).459C23, C4.4D4⋊75C22, C22⋊C4.88C23, (C2×Q8).281C23, C42.C2⋊50C22, C4○(C22.45C24), C42⋊C2⋊99C22, C22.19C24⋊18C2, C42⋊2C2⋊32C22, C22≀C2.24C22, (C23×C4).602C22, C22.45C24⋊27C2, C2.11(C2.C25), C4○(C22.47C24), C4○(C22.46C24), C4○(C22.50C24), C4○(C22.49C24), (C22×C4).1200C23, (C22×D4).594C22, C22.D4⋊45C22, C22.50C24⋊44C2, C23.37C23⋊31C2, C22.46C24⋊44C2, C23.36C23⋊21C2, C22.49C24⋊28C2, C22.47C24⋊43C2, (C2×C4×D4)⋊87C2, (C4×C4○D4)⋊21C2, C42⋊C2○(C4×D4), (C2×C4)⋊18(C4○D4), (C2×C4)○(D4⋊3Q8), C4.136(C2×C4○D4), (C2×C4⋊C4)⋊139C22, (C2×D4)○(C42⋊C2), C2.36(C22×C4○D4), C22.38(C2×C4○D4), (C2×C42⋊C2)⋊64C2, C22⋊C4○(C42⋊C2), (C2×C4○D4).324C22, (C2×C4)○(C22.45C24), (C2×C22⋊C4).544C22, (C2×C4)○(C22.49C24), (C2×C4)○(C22.46C24), SmallGroup(128,2207)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 788 in 566 conjugacy classes, 398 normal (50 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×21], C22, C22 [×6], C22 [×26], C2×C4 [×6], C2×C4 [×22], C2×C4 [×45], D4 [×4], D4 [×24], Q8 [×8], C23, C23 [×6], C23 [×10], C42 [×4], C42 [×18], C22⋊C4 [×42], C4⋊C4 [×4], C4⋊C4 [×38], C22×C4 [×3], C22×C4 [×24], C22×C4 [×8], C2×D4 [×3], C2×D4 [×12], C2×D4 [×4], C2×Q8, C2×Q8 [×4], C4○D4 [×12], C24 [×2], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×4], C42⋊C2, C42⋊C2 [×22], C4×D4 [×6], C4×D4 [×22], C4×Q8 [×2], C4×Q8 [×6], C22≀C2 [×4], C4⋊D4 [×10], C22⋊Q8 [×14], C22.D4 [×16], C4.4D4 [×6], C42.C2 [×8], C42⋊2C2 [×12], C4⋊Q8 [×2], C23×C4 [×2], C22×D4, C2×C4○D4, C2×C4○D4 [×2], C2×C42⋊C2 [×2], C2×C4×D4, C4×C4○D4, C4×C4○D4 [×2], C22.19C24 [×2], C23.36C23 [×6], C23.37C23, D4⋊5D4 [×2], D4⋊6D4, C22.45C24 [×4], C22.46C24 [×4], C22.47C24 [×2], D4⋊3Q8, C22.49C24, C22.50C24, C22.64C25
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], C25, C22×C4○D4 [×2], C2.C25, C22.64C25
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=b, g2=a, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >
►On 32 points
(1 15)(2 16)(3 13)(4 14)(5 20)(6 17)(7 18)(8 19)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)
(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 31)(2 27)(3 29)(4 25)(5 23)(6 9)(7 21)(8 11)(10 18)(12 20)(13 28)(14 30)(15 26)(16 32)(17 24)(19 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)
(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 3)(2 14)(4 16)(5 18)(6 8)(7 20)(9 11)(10 23)(12 21)(13 15)(17 19)(22 24)(25 32)(26 28)(27 30)(29 31)
(1 17 15 6)(2 18 16 7)(3 19 13 8)(4 20 14 5)(9 31 24 26)(10 32 21 27)(11 29 22 28)(12 30 23 25)
G:=sub<Sym(32)| (1,15)(2,16)(3,13)(4,14)(5,20)(6,17)(7,18)(8,19)(9,24)(10,21)(11,22)(12,23)(25,30)(26,31)(27,32)(28,29), (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), (1,31)(2,27)(3,29)(4,25)(5,23)(6,9)(7,21)(8,11)(10,18)(12,20)(13,28)(14,30)(15,26)(16,32)(17,24)(19,22), (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), (9,11)(10,12)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,3)(2,14)(4,16)(5,18)(6,8)(7,20)(9,11)(10,23)(12,21)(13,15)(17,19)(22,24)(25,32)(26,28)(27,30)(29,31), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,31,24,26)(10,32,21,27)(11,29,22,28)(12,30,23,25)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,20)(6,17)(7,18)(8,19)(9,24)(10,21)(11,22)(12,23)(25,30)(26,31)(27,32)(28,29), (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), (1,31)(2,27)(3,29)(4,25)(5,23)(6,9)(7,21)(8,11)(10,18)(12,20)(13,28)(14,30)(15,26)(16,32)(17,24)(19,22), (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), (9,11)(10,12)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,3)(2,14)(4,16)(5,18)(6,8)(7,20)(9,11)(10,23)(12,21)(13,15)(17,19)(22,24)(25,32)(26,28)(27,30)(29,31), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,31,24,26)(10,32,21,27)(11,29,22,28)(12,30,23,25) );
G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,20),(6,17),(7,18),(8,19),(9,24),(10,21),(11,22),(12,23),(25,30),(26,31),(27,32),(28,29)], [(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)], [(1,31),(2,27),(3,29),(4,25),(5,23),(6,9),(7,21),(8,11),(10,18),(12,20),(13,28),(14,30),(15,26),(16,32),(17,24),(19,22)], [(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)], [(9,11),(10,12),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,3),(2,14),(4,16),(5,18),(6,8),(7,20),(9,11),(10,23),(12,21),(13,15),(17,19),(22,24),(25,32),(26,28),(27,30),(29,31)], [(1,17,15,6),(2,18,16,7),(3,19,13,8),(4,20,14,5),(9,31,24,26),(10,32,21,27),(11,29,22,28),(12,30,23,25)])
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4V | 4W | ··· | 4AJ |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | C4○D4 | C2.C25 |
| kernel | C22.64C25 | C2×C42⋊C2 | C2×C4×D4 | C4×C4○D4 | C22.19C24 | C23.36C23 | C23.37C23 | D4⋊5D4 | D4⋊6D4 | C22.45C24 | C22.46C24 | C22.47C24 | D4⋊3Q8 | C22.49C24 | C22.50C24 | C2×C4 | D4 | C2 |
| # reps | 1 | 2 | 1 | 3 | 2 | 6 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2^2._{64}C_2^5 % in TeX
G:=Group("C2^2.64C2^5"); // GroupNames label
G:=SmallGroup(128,2207);
// by ID
G=gap.SmallGroup(128,2207);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,102]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,c*g=g*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations