direct product, metabelian, supersoluble, monomial
Aliases: C5×C22.F5, C102.2C4, Dic5.3C20, C52⋊11M4(2), C5⋊C8⋊2C10, C22.(C5×F5), (C2×C10).9F5, C2.6(C10×F5), C10.6(C2×C20), (C2×C10).2C20, C5⋊2(C5×M4(2)), C10.47(C2×F5), (C5×Dic5).11C4, (C2×Dic5).5C10, Dic5.8(C2×C10), (C10×Dic5).10C2, (C5×Dic5).13C22, (C5×C5⋊C8)⋊6C2, (C5×C10).18(C2×C4), SmallGroup(400,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.F5
G = < a,b,c,d,e | a5=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Smallest permutation representation of C5×C22.F5
►On 40 points
Generators in S40
(1 22 35 26 9)(2 23 36 27 10)(3 24 37 28 11)(4 17 38 29 12)(5 18 39 30 13)(6 19 40 31 14)(7 20 33 32 15)(8 21 34 25 16)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)
(1 35 9 22 26)(2 23 36 27 10)(3 28 24 11 37)(4 12 29 38 17)(5 39 13 18 30)(6 19 40 31 14)(7 32 20 15 33)(8 16 25 34 21)
(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)(33 34 35 36 37 38 39 40)
G:=sub<Sym(40)| (1,22,35,26,9)(2,23,36,27,10)(3,24,37,28,11)(4,17,38,29,12)(5,18,39,30,13)(6,19,40,31,14)(7,20,33,32,15)(8,21,34,25,16), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40), (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (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)(33,34,35,36,37,38,39,40)>;
G:=Group( (1,22,35,26,9)(2,23,36,27,10)(3,24,37,28,11)(4,17,38,29,12)(5,18,39,30,13)(6,19,40,31,14)(7,20,33,32,15)(8,21,34,25,16), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40), (1,35,9,22,26)(2,23,36,27,10)(3,28,24,11,37)(4,12,29,38,17)(5,39,13,18,30)(6,19,40,31,14)(7,32,20,15,33)(8,16,25,34,21), (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)(33,34,35,36,37,38,39,40) );
G=PermutationGroup([[(1,22,35,26,9),(2,23,36,27,10),(3,24,37,28,11),(4,17,38,29,12),(5,18,39,30,13),(6,19,40,31,14),(7,20,33,32,15),(8,21,34,25,16)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40)], [(1,35,9,22,26),(2,23,36,27,10),(3,28,24,11,37),(4,12,29,38,17),(5,39,13,18,30),(6,19,40,31,14),(7,32,20,15,33),(8,16,25,34,21)], [(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),(33,34,35,36,37,38,39,40)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10W | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | M4(2) | C5×M4(2) | F5 | C2×F5 | C22.F5 | C5×F5 | C10×F5 | C5×C22.F5 |
| kernel | C5×C22.F5 | C5×C5⋊C8 | C10×Dic5 | C5×Dic5 | C102 | C22.F5 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C52 | C5 | C2×C10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×C22.F5 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 16 | 0 | 0 | 0 |
| 7 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 33 | 37 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 15 | 0 | 0 |
| 0 | 9 | 0 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[16,7,0,0,0,18,0,0,0,0,10,33,0,0,0,37],[0,0,32,0,0,0,15,9,1,0,0,0,0,1,0,0] >;
C5×C22.F5 in GAP, Magma, Sage, TeX
C_5\times C_2^2.F_5
% in TeX
G:=Group("C5xC2^2.F5"); // GroupNames label
G:=SmallGroup(400,140);
// by ID
G=gap.SmallGroup(400,140);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,69,5765,599]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
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