p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.165C23, (C23×C4)⋊7C4, (C2×D4).260D4, (C22×D4)⋊16C4, C24.27(C2×C4), (C22×Q8)⋊13C4, C22.8C22≀C2, C23.122(C2×D4), (C22×C4).258D4, C2.7(C24⋊3C4), C23.34D4⋊6C2, C23.28(C22⋊C4), C23.181(C22×C4), (C23×C4).227C22, (C22×D4).449C22, C2.24(C23.C23), (C2×C23⋊C4)⋊2C2, (C22×C4).73(C2×C4), (C22×C4○D4).2C2, (C2×C4).41(C22⋊C4), (C2×C22⋊C4).4C22, C22.29(C2×C22⋊C4), SmallGroup(128,514)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.165C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=ade, fg=gf >
Subgroups: 644 in 306 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C2.C42, C23⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C23.34D4, C2×C23⋊C4, C22×C4○D4, C24.165C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, C23.C23, C24.165C23
►On 32 points
(1 3)(2 15)(4 13)(5 7)(6 11)(8 9)(10 12)(14 16)(17 23)(18 20)(19 21)(22 24)(25 27)(26 30)(28 32)(29 31)
(1 10)(2 11)(3 12)(4 9)(5 14)(6 15)(7 16)(8 13)(17 32)(18 29)(19 30)(20 31)(21 26)(22 27)(23 28)(24 25)
(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 16)(2 13)(3 14)(4 15)(5 12)(6 9)(7 10)(8 11)(17 21)(18 22)(19 23)(20 24)(25 31)(26 32)(27 29)(28 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 2)(3 15)(4 14)(5 9)(6 12)(7 8)(10 11)(13 16)(17 25)(18 30)(19 29)(20 26)(21 31)(22 28)(23 27)(24 32)
(1 22 16 18)(2 28 13 30)(3 24 14 20)(4 26 15 32)(5 31 12 25)(6 17 9 21)(7 29 10 27)(8 19 11 23)
G:=sub<Sym(32)| (1,3)(2,15)(4,13)(5,7)(6,11)(8,9)(10,12)(14,16)(17,23)(18,20)(19,21)(22,24)(25,27)(26,30)(28,32)(29,31), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13)(17,32)(18,29)(19,30)(20,31)(21,26)(22,27)(23,28)(24,25), (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), (1,16)(2,13)(3,14)(4,15)(5,12)(6,9)(7,10)(8,11)(17,21)(18,22)(19,23)(20,24)(25,31)(26,32)(27,29)(28,30), (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), (1,2)(3,15)(4,14)(5,9)(6,12)(7,8)(10,11)(13,16)(17,25)(18,30)(19,29)(20,26)(21,31)(22,28)(23,27)(24,32), (1,22,16,18)(2,28,13,30)(3,24,14,20)(4,26,15,32)(5,31,12,25)(6,17,9,21)(7,29,10,27)(8,19,11,23)>;
G:=Group( (1,3)(2,15)(4,13)(5,7)(6,11)(8,9)(10,12)(14,16)(17,23)(18,20)(19,21)(22,24)(25,27)(26,30)(28,32)(29,31), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13)(17,32)(18,29)(19,30)(20,31)(21,26)(22,27)(23,28)(24,25), (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), (1,16)(2,13)(3,14)(4,15)(5,12)(6,9)(7,10)(8,11)(17,21)(18,22)(19,23)(20,24)(25,31)(26,32)(27,29)(28,30), (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), (1,2)(3,15)(4,14)(5,9)(6,12)(7,8)(10,11)(13,16)(17,25)(18,30)(19,29)(20,26)(21,31)(22,28)(23,27)(24,32), (1,22,16,18)(2,28,13,30)(3,24,14,20)(4,26,15,32)(5,31,12,25)(6,17,9,21)(7,29,10,27)(8,19,11,23) );
G=PermutationGroup([[(1,3),(2,15),(4,13),(5,7),(6,11),(8,9),(10,12),(14,16),(17,23),(18,20),(19,21),(22,24),(25,27),(26,30),(28,32),(29,31)], [(1,10),(2,11),(3,12),(4,9),(5,14),(6,15),(7,16),(8,13),(17,32),(18,29),(19,30),(20,31),(21,26),(22,27),(23,28),(24,25)], [(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)], [(1,16),(2,13),(3,14),(4,15),(5,12),(6,9),(7,10),(8,11),(17,21),(18,22),(19,23),(20,24),(25,31),(26,32),(27,29),(28,30)], [(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)], [(1,2),(3,15),(4,14),(5,9),(6,12),(7,8),(10,11),(13,16),(17,25),(18,30),(19,29),(20,26),(21,31),(22,28),(23,27),(24,32)], [(1,22,16,18),(2,28,13,30),(3,24,14,20),(4,26,15,32),(5,31,12,25),(6,17,9,21),(7,29,10,27),(8,19,11,23)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C23.C23 |
| kernel | C24.165C23 | C23.34D4 | C2×C23⋊C4 | C22×C4○D4 | C23×C4 | C22×D4 | C22×Q8 | C22×C4 | C2×D4 | C2 |
| # reps | 1 | 2 | 4 | 1 | 4 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C24.165C23 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 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 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 4 |
| 0 | 0 | 4 | 1 | 4 | 0 |
| 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 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 4 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 0 | 4 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 1 | 2 | 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 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4,1,1,0,0,0,0,0,1,0,0,0,0,1,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,4,4,0,0,3,4,1,1,0,0,0,0,0,4,0,0,0,0,4,0],[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],[1,0,0,0,0,0,4,4,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,2,1,4,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,3,4,1,1,0,0,0,1,0,0],[3,1,0,0,0,0,2,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
C24.165C23 in GAP, Magma, Sage, TeX
C_2^4._{165}C_2^3 % in TeX
G:=Group("C2^4.165C2^3"); // GroupNames label
G:=SmallGroup(128,514);
// by ID
G=gap.SmallGroup(128,514);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,2028]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*c*f=c*d=d*c,c*e=e*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*e*f=a*d*e,f*g=g*f>;
// generators/relations