p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.93C24, C24.158C23, C42.135C23, C22.153C25, C22.142- 1+4, C4:Q8:47C22, D4:3Q8:45C2, (C4xD4):75C22, (C4xQ8):71C22, C4:C4.336C23, (C2xC4).143C24, C23:2Q8:10C2, C22:Q8:57C22, (C2xD4).341C23, C4.4D4:94C22, C22:C4.63C23, (C2xQ8).318C23, C42.C2:70C22, C42:C2:69C22, C42:2C2:49C22, C22wrC2.17C22, C4:D4.124C22, (C2xC42).978C22, (C22xC4).412C23, C22.32C24.4C2, C22.45C24:24C2, C2.56(C2x2- 1+4), C2.64(C2.C25), C22.D4:27C22, C22.36C24:41C2, C23.36C23:61C2, C22.57C24:20C2, C22.50C24:40C2, C22.46C24:40C2, C22.35C24:25C2, C22.33C24:24C2, C23.41C23:26C2, C23.37C23:57C2, (C2xC4:C4):94C22, (C2xC42:2C2):42C2, (C2xC22:C4).396C22, SmallGroup(128,2296)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.153C25
G = < a,b,c,d,e,f,g | a2=b2=d2=g2=1, c2=b, e2=a, f2=ba=ab, dcd=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 636 in 468 conjugacy classes, 380 normal (58 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, C2xC42, C2xC22:C4, C2xC22:C4, C2xC4:C4, C2xC4:C4, C42:C2, C4xD4, C4xD4, C4xQ8, C4xQ8, C22wrC2, C4:D4, C4:D4, C22:Q8, C22:Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42:2C2, C4:Q8, C4:Q8, C2xC42:2C2, C23.36C23, C23.37C23, C22.32C24, C22.33C24, C22.33C24, C22.35C24, C22.36C24, C23:2Q8, C23.41C23, C22.45C24, C22.46C24, D4:3Q8, C22.50C24, C22.57C24, C22.153C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2x2- 1+4, C2.C25, C22.153C25
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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 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)
(2 28)(4 26)(5 7)(6 19)(8 17)(9 13)(11 15)(18 20)(21 23)(22 32)(24 30)(29 31)
(1 31 27 23)(2 30 28 22)(3 29 25 21)(4 32 26 24)(5 12 20 16)(6 11 17 15)(7 10 18 14)(8 9 19 13)
(1 13 25 11)(2 16 26 10)(3 15 27 9)(4 14 28 12)(5 24 18 30)(6 23 19 29)(7 22 20 32)(8 21 17 31)
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (1,2,3,4)(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), (2,28)(4,26)(5,7)(6,19)(8,17)(9,13)(11,15)(18,20)(21,23)(22,32)(24,30)(29,31), (1,31,27,23)(2,30,28,22)(3,29,25,21)(4,32,26,24)(5,12,20,16)(6,11,17,15)(7,10,18,14)(8,9,19,13), (1,13,25,11)(2,16,26,10)(3,15,27,9)(4,14,28,12)(5,24,18,30)(6,23,19,29)(7,22,20,32)(8,21,17,31), (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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), (1,2,3,4)(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), (2,28)(4,26)(5,7)(6,19)(8,17)(9,13)(11,15)(18,20)(21,23)(22,32)(24,30)(29,31), (1,31,27,23)(2,30,28,22)(3,29,25,21)(4,32,26,24)(5,12,20,16)(6,11,17,15)(7,10,18,14)(8,9,19,13), (1,13,25,11)(2,16,26,10)(3,15,27,9)(4,14,28,12)(5,24,18,30)(6,23,19,29)(7,22,20,32)(8,21,17,31), (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32) );
G=PermutationGroup([[(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,15),(12,16),(21,29),(22,30),(23,31),(24,32)], [(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)], [(1,2,3,4),(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)], [(2,28),(4,26),(5,7),(6,19),(8,17),(9,13),(11,15),(18,20),(21,23),(22,32),(24,30),(29,31)], [(1,31,27,23),(2,30,28,22),(3,29,25,21),(4,32,26,24),(5,12,20,16),(6,11,17,15),(7,10,18,14),(8,9,19,13)], [(1,13,25,11),(2,16,26,10),(3,15,27,9),(4,14,28,12),(5,24,18,30),(6,23,19,29),(7,22,20,32),(8,21,17,31)], [(2,28),(4,26),(5,20),(7,18),(10,14),(12,16),(22,30),(24,32)]])
38 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4AB |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
38 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | 2- 1+4 | C2.C25 |
kernel | C22.153C25 | C2xC42:2C2 | C23.36C23 | C23.37C23 | C22.32C24 | C22.33C24 | C22.35C24 | C22.36C24 | C23:2Q8 | C23.41C23 | C22.45C24 | C22.46C24 | D4:3Q8 | C22.50C24 | C22.57C24 | C22 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 2 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C22.153C25 ►in GL8(F5)
4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 1 | 0 | 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 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
C22.153C25 in GAP, Magma, Sage, TeX
C_2^2._{153}C_2^5
% in TeX
G:=Group("C2^2.153C2^5");
// GroupNames label
G:=SmallGroup(128,2296);
// by ID
G=gap.SmallGroup(128,2296);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,723,184,2019,570,360,1684]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=g^2=1,c^2=b,e^2=a,f^2=b*a=a*b,d*c*d=g*c*g=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations