p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.33C25, C24.611C23, C23.114C24, C42.539C23, C42○(C4⋊Q8), C42○C22≀C2, (C4×D4)⋊92C22, C42○(C4⋊1D4), C42○(C4⋊D4), (C2×C4).36C24, C4⋊Q8⋊107C22, (C4×Q8)⋊86C22, C42○(C22⋊Q8), C4⋊1D4⋊58C22, C4⋊C4.460C23, (C22×C42)⋊25C2, (C2×C42)⋊43C22, C42○(C4.4D4), C42○(C42.C2), C42○(C42⋊2C2), (C2×D4).446C23, C4.4D4⋊97C22, C22⋊C4.74C23, (C2×Q8).420C23, C42.C2⋊74C22, C4○2(C22.19C24), C42⋊2C2⋊52C22, C42⋊C2⋊85C22, C22.19C24⋊45C2, C22≀C2.34C22, C4⋊D4.240C22, (C23×C4).708C22, C42○(C22.D4), C22⋊Q8.240C22, C4○2(C22.26C24), C22.26C24⋊58C2, (C22×C4).1297C23, C42○(C22.19C24), C4○2(C23.37C23), C4○2(C23.36C23), C23.36C23⋊66C2, C23.37C23⋊59C2, C42○(C22.26C24), C22.D4.41C22, C42○(C23.37C23), C42○(C23.36C23), (C4×C4○D4)⋊16C2, C4.71(C2×C4○D4), (C2×C4)⋊15(C4○D4), C22.7(C2×C4○D4), C2.15(C22×C4○D4), (C2×C4○D4).320C22, SmallGroup(128,2176)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 812 in 606 conjugacy classes, 408 normal (8 characteristic)
C1, C2 [×3], C2 [×10], C4 [×12], C4 [×18], C22, C22 [×6], C22 [×26], C2×C4 [×36], C2×C4 [×54], D4 [×36], Q8 [×12], C23 [×7], C23 [×6], C42 [×2], C42 [×26], C22⋊C4 [×36], C4⋊C4 [×36], C22×C4 [×30], C22×C4 [×12], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×24], C24, C2×C42 [×16], C42⋊C2 [×18], C4×D4 [×36], C4×Q8 [×12], C22≀C2 [×4], C4⋊D4 [×12], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4 [×6], C42.C2 [×6], C42⋊2C2 [×8], C4⋊1D4, C4⋊Q8 [×3], C23×C4 [×3], C2×C4○D4 [×6], C22×C42, C4×C4○D4 [×6], C22.19C24 [×6], C23.36C23 [×12], C22.26C24 [×3], C23.37C23 [×3], C22.33C25
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×12], C24 [×31], C2×C4○D4 [×18], C25, C22×C4○D4 [×3], C22.33C25
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=b, g2=a, ab=ba, dcd=ac=ca, 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, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >
►On 32 points
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 22)(10 23)(11 24)(12 21)(25 31)(26 32)(27 29)(28 30)
(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 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)
(1 3)(2 4)(5 7)(6 8)(9 24)(10 21)(11 22)(12 23)(13 15)(14 16)(17 19)(18 20)(25 29)(26 30)(27 31)(28 32)
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 24)(10 21)(11 22)(12 23)(25 29)(26 30)(27 31)(28 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 19 15 6)(2 20 16 7)(3 17 13 8)(4 18 14 5)(9 31 22 25)(10 32 23 26)(11 29 24 27)(12 30 21 28)
G:=sub<Sym(32)| (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(25,31)(26,32)(27,29)(28,30), (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), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (1,3)(2,4)(5,7)(6,8)(9,24)(10,21)(11,22)(12,23)(13,15)(14,16)(17,19)(18,20)(25,29)(26,30)(27,31)(28,32), (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,24)(10,21)(11,22)(12,23)(25,29)(26,30)(27,31)(28,32), (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), (1,19,15,6)(2,20,16,7)(3,17,13,8)(4,18,14,5)(9,31,22,25)(10,32,23,26)(11,29,24,27)(12,30,21,28)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(25,31)(26,32)(27,29)(28,30), (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), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32), (1,3)(2,4)(5,7)(6,8)(9,24)(10,21)(11,22)(12,23)(13,15)(14,16)(17,19)(18,20)(25,29)(26,30)(27,31)(28,32), (1,15)(2,16)(3,13)(4,14)(5,18)(6,19)(7,20)(8,17)(9,24)(10,21)(11,22)(12,23)(25,29)(26,30)(27,31)(28,32), (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), (1,19,15,6)(2,20,16,7)(3,17,13,8)(4,18,14,5)(9,31,22,25)(10,32,23,26)(11,29,24,27)(12,30,21,28) );
G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,18),(6,19),(7,20),(8,17),(9,22),(10,23),(11,24),(12,21),(25,31),(26,32),(27,29),(28,30)], [(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)], [(1,25),(2,26),(3,27),(4,28),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)], [(1,3),(2,4),(5,7),(6,8),(9,24),(10,21),(11,22),(12,23),(13,15),(14,16),(17,19),(18,20),(25,29),(26,30),(27,31),(28,32)], [(1,15),(2,16),(3,13),(4,14),(5,18),(6,19),(7,20),(8,17),(9,24),(10,21),(11,22),(12,23),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,19,15,6),(2,20,16,7),(3,17,13,8),(4,18,14,5),(9,31,22,25),(10,32,23,26),(11,29,24,27),(12,30,21,28)])
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 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
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],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4L | 4M | ··· | 4AD | 4AE | ··· | 4AP |
| 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 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 |
| kernel | C22.33C25 | C22×C42 | C4×C4○D4 | C22.19C24 | C23.36C23 | C22.26C24 | C23.37C23 | C2×C4 |
| # reps | 1 | 1 | 6 | 6 | 12 | 3 | 3 | 24 |
In GAP, Magma, Sage, TeX
C_2^2._{33}C_2^5 % in TeX
G:=Group("C2^2.33C2^5"); // GroupNames label
G:=SmallGroup(128,2176);
// by ID
G=gap.SmallGroup(128,2176);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,248,102]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=b,g^2=a,a*b=b*a,d*c*d=a*c=c*a,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*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations