p-group, metabelian, nilpotent (class 2), monomial
Aliases: C25.58C22, C23.584C24, C24.391C23, C22.3582+ 1+4, (C2×D4).138D4, C23.63(C2×D4), C24⋊3C4⋊24C2, (C23×C4)⋊12C22, (C2×C42)⋊31C22, C23⋊Q8⋊41C2, C2.89(D4⋊5D4), (C22×Q8)⋊8C22, C23.168(C4○D4), C23.23D4⋊83C2, C23.10D4⋊79C2, C23.11D4⋊77C2, C2.42(C23⋊3D4), (C22×C4).179C23, C22.393(C22×D4), C2.C42⋊37C22, C2.5(C24⋊C22), (C22×D4).223C22, C24.C22⋊122C2, C2.61(C22.32C24), C2.75(C22.45C24), (C2×C4⋊C4)⋊33C22, (C2×C4).416(C2×D4), (C2×C4.4D4)⋊27C2, (C2×C22≀C2).13C2, (C2×C22⋊C4)⋊30C22, C22.446(C2×C4○D4), SmallGroup(128,1416)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.584C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >
Subgroups: 836 in 344 conjugacy classes, 96 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22≀C2, C4.4D4, C23×C4, C22×D4, C22×D4, C22×Q8, C25, C24⋊3C4, C23.23D4, C23.23D4, C24.C22, C23⋊Q8, C23.10D4, C23.11D4, C2×C22≀C2, C2×C4.4D4, C23.584C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C23⋊3D4, C22.32C24, D4⋊5D4, C22.45C24, C24⋊C22, C23.584C24
(1 7)(2 8)(3 5)(4 6)(9 30)(10 31)(11 32)(12 29)(13 21)(14 22)(15 23)(16 24)(17 27)(18 28)(19 25)(20 26)
(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 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 25)(12 26)(17 30)(18 31)(19 32)(20 29)
(1 28)(2 19)(3 26)(4 17)(5 20)(6 27)(7 18)(8 25)(9 22)(10 15)(11 24)(12 13)(14 30)(16 32)(21 29)(23 31)
(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 13)(2 4)(3 15)(5 23)(6 8)(7 21)(10 28)(12 26)(14 16)(18 31)(20 29)(22 24)
(1 15)(2 24)(3 13)(4 22)(5 21)(6 14)(7 23)(8 16)(9 25)(10 20)(11 27)(12 18)(17 32)(19 30)(26 31)(28 29)
G:=sub<Sym(32)| (1,7)(2,8)(3,5)(4,6)(9,30)(10,31)(11,32)(12,29)(13,21)(14,22)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26), (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), (1,15)(2,16)(3,13)(4,14)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,25)(12,26)(17,30)(18,31)(19,32)(20,29), (1,28)(2,19)(3,26)(4,17)(5,20)(6,27)(7,18)(8,25)(9,22)(10,15)(11,24)(12,13)(14,30)(16,32)(21,29)(23,31), (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), (1,13)(2,4)(3,15)(5,23)(6,8)(7,21)(10,28)(12,26)(14,16)(18,31)(20,29)(22,24), (1,15)(2,24)(3,13)(4,22)(5,21)(6,14)(7,23)(8,16)(9,25)(10,20)(11,27)(12,18)(17,32)(19,30)(26,31)(28,29)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,30)(10,31)(11,32)(12,29)(13,21)(14,22)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26), (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), (1,15)(2,16)(3,13)(4,14)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,25)(12,26)(17,30)(18,31)(19,32)(20,29), (1,28)(2,19)(3,26)(4,17)(5,20)(6,27)(7,18)(8,25)(9,22)(10,15)(11,24)(12,13)(14,30)(16,32)(21,29)(23,31), (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), (1,13)(2,4)(3,15)(5,23)(6,8)(7,21)(10,28)(12,26)(14,16)(18,31)(20,29)(22,24), (1,15)(2,24)(3,13)(4,22)(5,21)(6,14)(7,23)(8,16)(9,25)(10,20)(11,27)(12,18)(17,32)(19,30)(26,31)(28,29) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,30),(10,31),(11,32),(12,29),(13,21),(14,22),(15,23),(16,24),(17,27),(18,28),(19,25),(20,26)], [(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)], [(1,15),(2,16),(3,13),(4,14),(5,21),(6,22),(7,23),(8,24),(9,27),(10,28),(11,25),(12,26),(17,30),(18,31),(19,32),(20,29)], [(1,28),(2,19),(3,26),(4,17),(5,20),(6,27),(7,18),(8,25),(9,22),(10,15),(11,24),(12,13),(14,30),(16,32),(21,29),(23,31)], [(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)], [(1,13),(2,4),(3,15),(5,23),(6,8),(7,21),(10,28),(12,26),(14,16),(18,31),(20,29),(22,24)], [(1,15),(2,24),(3,13),(4,22),(5,21),(6,14),(7,23),(8,16),(9,25),(10,20),(11,27),(12,18),(17,32),(19,30),(26,31),(28,29)]])
32 conjugacy classes
class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4J | 4K | ··· | 4P |
order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 |
kernel | C23.584C24 | C24⋊3C4 | C23.23D4 | C24.C22 | C23⋊Q8 | C23.10D4 | C23.11D4 | C2×C22≀C2 | C2×C4.4D4 | C2×D4 | C23 | C22 |
# reps | 1 | 2 | 3 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 4 |
Matrix representation of C23.584C24 ►in GL6(𝔽5)
1 | 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 | 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 |
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 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 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 | 4 |
3 | 0 | 0 | 0 | 0 | 0 |
0 | 3 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 2 |
0 | 0 | 0 | 0 | 1 | 2 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 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 |
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 | 1 | 0 |
0 | 0 | 0 | 0 | 2 | 4 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,4,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,1,0,0,0,0,2,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,0,4] >;
C23.584C24 in GAP, Magma, Sage, TeX
C_2^3._{584}C_2^4
% in TeX
G:=Group("C2^3.584C2^4");
// GroupNames label
G:=SmallGroup(128,1416);
// by ID
G=gap.SmallGroup(128,1416);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,1571,346]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*g=g*f>;
// generators/relations