p-group, metabelian, nilpotent (class 2), monomial
Aliases: C25.23C22, C24.551C23, C23.224C24, C22.612+ (1+4), C24.70(C2×C4), (C2×C42)⋊4C22, C22.65(C4×D4), C2.3(D4⋊5D4), C24⋊3C4.5C2, C22⋊C4.182D4, C23.413(C2×D4), C23.7Q8⋊22C2, C23.8Q8⋊12C2, C22⋊2(C42⋊C2), C23.287(C4○D4), C23.34D4⋊15C2, (C23×C4).298C22, C23.215(C22×C4), C22.115(C23×C4), (C22×C4).753C23, C22.103(C22×D4), C2.C42⋊52C22, C24.C22⋊12C2, C2.1(C22.45C24), C2.24(C22.11C24), C2.29(C2×C4×D4), (C2×C4⋊C4)⋊8C22, (C4×C22⋊C4)⋊8C2, C22⋊C4⋊39(C2×C4), (C2×C22⋊C4)⋊23C4, (C22×C4)⋊30(C2×C4), (C2×C4).884(C2×D4), C22⋊C4○3(C22⋊C4), (C2×C42⋊C2)⋊11C2, (C2×C4).229(C22×C4), C2.25(C2×C42⋊C2), C22.109(C2×C4○D4), (C22×C22⋊C4).12C2, (C2×C22⋊C4).435C22, C22⋊C4○2(C2×C22⋊C4), SmallGroup(128,1074)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 780 in 396 conjugacy classes, 156 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×18], C22 [×3], C22 [×12], C22 [×50], C2×C4 [×12], C2×C4 [×50], C23, C23 [×14], C23 [×50], C42 [×6], C22⋊C4 [×20], C22⋊C4 [×16], C4⋊C4 [×6], C22×C4 [×4], C22×C4 [×16], C22×C4 [×16], C24 [×3], C24 [×4], C24 [×10], C2.C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×8], C2×C22⋊C4 [×14], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C42⋊C2 [×4], C23×C4 [×2], C23×C4 [×2], C25, C4×C22⋊C4, C4×C22⋊C4 [×2], C24⋊3C4, C23.7Q8, C23.34D4, C23.8Q8 [×2], C24.C22 [×4], C22×C22⋊C4 [×2], C2×C42⋊C2, C23.224C24
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×6], C24, C42⋊C2 [×4], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C42⋊C2, C2×C4×D4, C22.11C24, D4⋊5D4 [×2], C22.45C24 [×2], C23.224C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, 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, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(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)
(1 31 27 6)(2 18 28 24)(3 29 25 8)(4 20 26 22)(5 12 30 14)(7 10 32 16)(9 17 15 23)(11 19 13 21)
(1 3)(2 26)(4 28)(5 32)(6 8)(7 30)(9 11)(10 14)(12 16)(13 15)(17 19)(18 22)(20 24)(21 23)(25 27)(29 31)
(1 25)(2 26)(3 27)(4 28)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 31)(22 32)(23 29)(24 30)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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), (1,31,27,6)(2,18,28,24)(3,29,25,8)(4,20,26,22)(5,12,30,14)(7,10,32,16)(9,17,15,23)(11,19,13,21), (1,3)(2,26)(4,28)(5,32)(6,8)(7,30)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27)(29,31), (1,25)(2,26)(3,27)(4,28)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,31)(22,32)(23,29)(24,30)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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), (1,31,27,6)(2,18,28,24)(3,29,25,8)(4,20,26,22)(5,12,30,14)(7,10,32,16)(9,17,15,23)(11,19,13,21), (1,3)(2,26)(4,28)(5,32)(6,8)(7,30)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27)(29,31), (1,25)(2,26)(3,27)(4,28)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,31)(22,32)(23,29)(24,30) );
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)], [(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(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)], [(1,31,27,6),(2,18,28,24),(3,29,25,8),(4,20,26,22),(5,12,30,14),(7,10,32,16),(9,17,15,23),(11,19,13,21)], [(1,3),(2,26),(4,28),(5,32),(6,8),(7,30),(9,11),(10,14),(12,16),(13,15),(17,19),(18,22),(20,24),(21,23),(25,27),(29,31)], [(1,25),(2,26),(3,27),(4,28),(5,18),(6,19),(7,20),(8,17),(9,13),(10,14),(11,15),(12,16),(21,31),(22,32),(23,29),(24,30)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 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 | 0 | 1 |
| 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 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,1,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 2P | 2Q | 4A | ··· | 4P | 4Q | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | C4○D4 | 2+ (1+4) |
| kernel | C23.224C24 | C4×C22⋊C4 | C24⋊3C4 | C23.7Q8 | C23.34D4 | C23.8Q8 | C24.C22 | C22×C22⋊C4 | C2×C42⋊C2 | C2×C22⋊C4 | C22⋊C4 | C23 | C22 |
| # reps | 1 | 3 | 1 | 1 | 1 | 2 | 4 | 2 | 1 | 16 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_2^3._{224}C_2^4 % in TeX
G:=Group("C2^3.224C2^4"); // GroupNames label
G:=SmallGroup(128,1074);
// by ID
G=gap.SmallGroup(128,1074);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,346]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c,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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations