p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.89C25, C23.45C24, C42.577C23, C24.136C23, C4oD4:12D4, D4.60(C2xD4), D4o(C4.4D4), C4:Q8:91C22, Q8.62(C2xD4), Q8o(C4.4D4), D4:5D4:19C2, Q8:5D4:17C2, (C4xD4):45C22, (C2xC4).79C24, C2.35(D4xC23), C22wrC2:9C22, C4:C4.295C23, C4:1D4:52C22, C4:D4:28C22, (C2xC42):60C22, (C4xQ8):100C22, C4.124(C22xD4), C22:Q8:32C22, (C2xD4).472C23, C4.4D4:83C22, C22:C4.24C23, (C2xQ8).449C23, (C22xQ8):32C22, C22.16(C22xD4), C22.29C24:23C2, (C22xC4).361C23, (C2x2+ 1+4):12C2, (C2x2- 1+4):10C2, C22.D4:7C22, C42:C2:101C22, C2.24(C2.C25), C22.26C24:38C2, (C22xD4).426C22, C23.38C23:23C2, (C4xC4oD4):28C2, (C2xC4).189(C2xD4), (C2xC4.4D4):54C2, (C2xC4oD4):31C22, (C2xC22:C4):49C22, SmallGroup(128,2232)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.89C25
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=ba=ab, g2=a, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 1260 in 770 conjugacy classes, 428 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C24, C2xC42, C2xC22:C4, C42:C2, C4xD4, C4xQ8, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C4.4D4, C4:1D4, C4:Q8, C22xD4, C22xQ8, C2xC4oD4, C2xC4oD4, 2+ 1+4, 2- 1+4, C4xC4oD4, C2xC4.4D4, C22.26C24, C22.29C24, C23.38C23, D4:5D4, Q8:5D4, C2x2+ 1+4, C2x2- 1+4, C22.89C25
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, C25, D4xC23, C2.C25, C22.89C25
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(1 15)(2 24)(3 13)(4 22)(5 21)(6 14)(7 23)(8 16)(9 25)(10 18)(11 27)(12 20)(17 29)(19 31)(26 30)(28 32)
(1 19)(2 20)(3 17)(4 18)(5 25)(6 26)(7 27)(8 28)(9 15)(10 16)(11 13)(12 14)(21 31)(22 32)(23 29)(24 30)
(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 31)(2 32)(3 29)(4 30)(5 9)(6 10)(7 11)(8 12)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)
(1 21 5 15)(2 22 6 16)(3 23 7 13)(4 24 8 14)(9 19 31 25)(10 20 32 26)(11 17 29 27)(12 18 30 28)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,15)(2,24)(3,13)(4,22)(5,21)(6,14)(7,23)(8,16)(9,25)(10,18)(11,27)(12,20)(17,29)(19,31)(26,30)(28,32), (1,19)(2,20)(3,17)(4,18)(5,25)(6,26)(7,27)(8,28)(9,15)(10,16)(11,13)(12,14)(21,31)(22,32)(23,29)(24,30), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (1,21,5,15)(2,22,6,16)(3,23,7,13)(4,24,8,14)(9,19,31,25)(10,20,32,26)(11,17,29,27)(12,18,30,28)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,15)(2,24)(3,13)(4,22)(5,21)(6,14)(7,23)(8,16)(9,25)(10,18)(11,27)(12,20)(17,29)(19,31)(26,30)(28,32), (1,19)(2,20)(3,17)(4,18)(5,25)(6,26)(7,27)(8,28)(9,15)(10,16)(11,13)(12,14)(21,31)(22,32)(23,29)(24,30), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (1,21,5,15)(2,22,6,16)(3,23,7,13)(4,24,8,14)(9,19,31,25)(10,20,32,26)(11,17,29,27)(12,18,30,28) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,31),(10,32),(11,29),(12,30),(13,23),(14,24),(15,21),(16,22),(17,27),(18,28),(19,25),(20,26)], [(1,7),(2,8),(3,5),(4,6),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28)], [(1,15),(2,24),(3,13),(4,22),(5,21),(6,14),(7,23),(8,16),(9,25),(10,18),(11,27),(12,20),(17,29),(19,31),(26,30),(28,32)], [(1,19),(2,20),(3,17),(4,18),(5,25),(6,26),(7,27),(8,28),(9,15),(10,16),(11,13),(12,14),(21,31),(22,32),(23,29),(24,30)], [(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)], [(1,31),(2,32),(3,29),(4,30),(5,9),(6,10),(7,11),(8,12),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28)], [(1,21,5,15),(2,22,6,16),(3,23,7,13),(4,24,8,14),(9,19,31,25),(10,20,32,26),(11,17,29,27),(12,18,30,28)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | ··· | 2Q | 4A | ··· | 4L | 4M | ··· | 4Z |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C2.C25 |
kernel | C22.89C25 | C4xC4oD4 | C2xC4.4D4 | C22.26C24 | C22.29C24 | C23.38C23 | D4:5D4 | Q8:5D4 | C2x2+ 1+4 | C2x2- 1+4 | C4oD4 | C2 |
# reps | 1 | 1 | 3 | 3 | 3 | 3 | 12 | 4 | 1 | 1 | 8 | 4 |
Matrix representation of C22.89C25 ►in GL6(F5)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 2 | 0 | 0 | 0 |
0 | 0 | 0 | 2 | 0 | 0 |
0 | 0 | 0 | 0 | 2 | 0 |
0 | 0 | 0 | 0 | 0 | 2 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
C22.89C25 in GAP, Magma, Sage, TeX
C_2^2._{89}C_2^5
% in TeX
G:=Group("C2^2.89C2^5");
// GroupNames label
G:=SmallGroup(128,2232);
// by ID
G=gap.SmallGroup(128,2232);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,520,570,1684,102]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b*a=a*b,g^2=a,d*c*d=g*c*g^-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^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations