p-group, metabelian, nilpotent (class 3), monomial
Aliases: 2- 1+4⋊5C4, 2+ 1+4⋊6C4, C42.671C23, M4(2).26C23, D4○C4≀C2, Q8○C4≀C2, C4○D4.57D4, C4≀C2⋊16C22, (C2×D4).291D4, C4.12(C23×C4), (C2×Q8).226D4, Q8○M4(2)⋊13C2, (C2×C4).182C24, (C2×C42)⋊35C22, C4○D4.21C23, D4.23(C22×C4), C4.182(C22×D4), C23.229(C2×D4), Q8.23(C22×C4), D4.21(C22⋊C4), C2.C25.3C2, Q8.21(C22⋊C4), C22.29(C22×D4), C42⋊C2⋊77C22, C42⋊C22⋊19C2, (C2×M4(2))⋊43C22, (C22×C4).904C23, C4○D4○C4≀C2, C4○D4⋊3(C2×C4), (C4×C4○D4)⋊2C2, (C2×C4≀C2)⋊30C2, (C2×D4)⋊25(C2×C4), (C2×Q8)⋊19(C2×C4), (C2×C4).448(C2×D4), C4.35(C2×C22⋊C4), (C2×C4).64(C22×C4), C22.8(C2×C22⋊C4), (C2×C4○D4).86C22, C2.44(C22×C22⋊C4), SmallGroup(128,1633)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 2- 1+4⋊5C4
G = < a,b,c,d,e | a4=b2=d2=e4=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede-1=a2cd >
Subgroups: 636 in 369 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4≀C2, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C8○D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, C2×C4≀C2, C42⋊C22, C4×C4○D4, Q8○M4(2), C2.C25, 2- 1+4⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4, 2- 1+4⋊5C4
►On 16 points - transitive group 16T207
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(5 7)(10 12)(13 15)
(1 10 3 12)(2 11 4 9)(5 15 7 13)(6 16 8 14)
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)
(1 10 3 12)(2 11 4 9)(5 7)(6 8)(13 15)(14 16)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,7)(10,12)(13,15), (1,10,3,12)(2,11,4,9)(5,15,7,13)(6,16,8,14), (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15), (1,10,3,12)(2,11,4,9)(5,7)(6,8)(13,15)(14,16)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(5,7)(10,12)(13,15), (1,10,3,12)(2,11,4,9)(5,15,7,13)(6,16,8,14), (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15), (1,10,3,12)(2,11,4,9)(5,7)(6,8)(13,15)(14,16) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(5,7),(10,12),(13,15)], [(1,10,3,12),(2,11,4,9),(5,15,7,13),(6,16,8,14)], [(1,5),(2,6),(3,7),(4,8),(9,16),(10,13),(11,14),(12,15)], [(1,10,3,12),(2,11,4,9),(5,7),(6,8),(13,15),(14,16)]])
G:=TransitiveGroup(16,207);
44 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | ··· | 4M | 4N | ··· | 4W | 8A | ··· | 8H |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | 2- 1+4⋊5C4 |
| kernel | 2- 1+4⋊5C4 | C2×C4≀C2 | C42⋊C22 | C4×C4○D4 | Q8○M4(2) | C2.C25 | 2+ 1+4 | 2- 1+4 | C2×D4 | C2×Q8 | C4○D4 | C1 |
| # reps | 1 | 6 | 6 | 1 | 1 | 1 | 8 | 8 | 3 | 1 | 4 | 4 |
Matrix representation of 2- 1+4⋊5C4 ►in GL4(𝔽5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,4,0,0,4,0,0,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,0,3,0,0,0,0,1,2,0,0,0,0,1,0,0],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1] >;
2- 1+4⋊5C4 in GAP, Magma, Sage, TeX
2_-^{1+4}\rtimes_5C_4 % in TeX
G:=Group("ES-(2,2):5C4"); // GroupNames label
G:=SmallGroup(128,1633);
// by ID
G=gap.SmallGroup(128,1633);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=d^2=e^4=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e^-1=a^2*c*d>;
// generators/relations