direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4×D52, C20⋊4D10, Dic5⋊5D10, D10.16D10, (D5×C20)⋊8C2, (C5×C20)⋊5C22, (D5×Dic5)⋊6C2, C52⋊3(C22×C4), (C5×C10).7C23, C10.7(C22×D5), Dic5⋊2D5⋊5C2, C52⋊6C4⋊2C22, (C5×Dic5)⋊5C22, (D5×C10).12C22, C5⋊2(C2×C4×D5), C2.1(C2×D52), (C4×C5⋊D5)⋊7C2, C5⋊D5⋊3(C2×C4), (C2×D52).6C2, (C5×D5)⋊3(C2×C4), (C2×C5⋊D5).15C22, SmallGroup(400,169)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C4×D52 |
Generators and relations for C4×D52
G = < a,b,c,d,e | a4=b5=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 748 in 124 conjugacy classes, 46 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C5×D5, C5⋊D5, C5×C10, C2×C4×D5, C5×Dic5, C52⋊6C4, C5×C20, D52, D5×C10, C2×C5⋊D5, D5×Dic5, Dic5⋊2D5, D5×C20, C4×C5⋊D5, C2×D52, C4×D52
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C4×D5, C22×D5, C2×C4×D5, D52, C2×D52, C4×D52
►On 40 points
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(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)
(1 21)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 26)(9 30)(10 29)(11 33)(12 32)(13 31)(14 35)(15 34)(16 38)(17 37)(18 36)(19 40)(20 39)
(1 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(16 18 20 17 19)(21 24 22 25 23)(26 29 27 30 28)(31 34 32 35 33)(36 39 37 40 38)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 22)(7 23)(8 24)(9 25)(10 21)(11 37)(12 38)(13 39)(14 40)(15 36)(16 32)(17 33)(18 34)(19 35)(20 31)
G:=sub<Sym(40)| (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (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), (1,21)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,26)(9,30)(10,29)(11,33)(12,32)(13,31)(14,35)(15,34)(16,38)(17,37)(18,36)(19,40)(20,39), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(16,18,20,17,19)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)(36,39,37,40,38), (1,29)(2,30)(3,26)(4,27)(5,28)(6,22)(7,23)(8,24)(9,25)(10,21)(11,37)(12,38)(13,39)(14,40)(15,36)(16,32)(17,33)(18,34)(19,35)(20,31)>;
G:=Group( (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (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), (1,21)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,26)(9,30)(10,29)(11,33)(12,32)(13,31)(14,35)(15,34)(16,38)(17,37)(18,36)(19,40)(20,39), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(16,18,20,17,19)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)(36,39,37,40,38), (1,29)(2,30)(3,26)(4,27)(5,28)(6,22)(7,23)(8,24)(9,25)(10,21)(11,37)(12,38)(13,39)(14,40)(15,36)(16,32)(17,33)(18,34)(19,35)(20,31) );
G=PermutationGroup([[(1,18,8,13),(2,19,9,14),(3,20,10,15),(4,16,6,11),(5,17,7,12),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(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)], [(1,21),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,26),(9,30),(10,29),(11,33),(12,32),(13,31),(14,35),(15,34),(16,38),(17,37),(18,36),(19,40),(20,39)], [(1,3,5,2,4),(6,8,10,7,9),(11,13,15,12,14),(16,18,20,17,19),(21,24,22,25,23),(26,29,27,30,28),(31,34,32,35,33),(36,39,37,40,38)], [(1,29),(2,30),(3,26),(4,27),(5,28),(6,22),(7,23),(8,24),(9,25),(10,21),(11,37),(12,38),(13,39),(14,40),(15,36),(16,32),(17,33),(18,34),(19,35),(20,31)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 25 | 25 | 1 | 1 | 5 | 5 | 5 | 5 | 25 | 25 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | D52 | C2×D52 | C4×D52 |
| kernel | C4×D52 | D5×Dic5 | Dic5⋊2D5 | D5×C20 | C4×C5⋊D5 | C2×D52 | D52 | C4×D5 | Dic5 | C20 | D10 | D5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 4 | 4 | 4 | 16 | 4 | 4 | 8 |
Matrix representation of C4×D52 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 40 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 35 | 35 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,6,35,0,0,40,35] >;
C4×D52 in GAP, Magma, Sage, TeX
C_4\times D_5^2
% in TeX
G:=Group("C4xD5^2"); // GroupNames label
G:=SmallGroup(400,169);
// by ID
G=gap.SmallGroup(400,169);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^5=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations