direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4xC52:C4, C20:4F5, C52:7C42, C5:3(C4xF5), (C5xC20):8C4, C52:6C4:8C4, C10.23(C2xF5), C5:D5.9(C2xC4), (C4xC5:D5).13C2, C2.2(C2xC52:C4), (C5xC10).36(C2xC4), (C2xC52:C4).7C2, (C2xC5:D5).24C22, SmallGroup(400,158)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C4xC52:C4 |
Generators and relations for C4xC52:C4
G = < a,b,c,d | a4=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >
Subgroups: 556 in 76 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2xC4, D5, C10, C10, C42, Dic5, C20, C20, F5, D10, C52, C4xD5, C2xF5, C5:D5, C5xC10, C4xF5, C52:6C4, C5xC20, C52:C4, C2xC5:D5, C4xC5:D5, C2xC52:C4, C4xC52:C4
Quotients: C1, C2, C4, C22, C2xC4, C42, F5, C2xF5, C4xF5, C52:C4, C2xC52:C4, C4xC52:C4
►On 40 points
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(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 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 33)(2 31 5 35)(3 34 4 32)(6 39 7 37)(8 40 10 36)(9 38)(11 29 12 27)(13 30 15 26)(14 28)(16 24 17 22)(18 25 20 21)(19 23)
G:=sub<Sym(40)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(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), (1,3,5,2,4)(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), (1,33)(2,31,5,35)(3,34,4,32)(6,39,7,37)(8,40,10,36)(9,38)(11,29,12,27)(13,30,15,26)(14,28)(16,24,17,22)(18,25,20,21)(19,23)>;
G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(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), (1,3,5,2,4)(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), (1,33)(2,31,5,35)(3,34,4,32)(6,39,7,37)(8,40,10,36)(9,38)(11,29,12,27)(13,30,15,26)(14,28)(16,24,17,22)(18,25,20,21)(19,23) );
G=PermutationGroup([[(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(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)], [(1,3,5,2,4),(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)], [(1,33),(2,31,5,35),(3,34,4,32),(6,39,7,37),(8,40,10,36),(9,38),(11,29,12,27),(13,30,15,26),(14,28),(16,24,17,22),(18,25,20,21),(19,23)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4L | 5A | ··· | 5F | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 25 | ··· | 25 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | F5 | C2xF5 | C4xF5 | C52:C4 | C2xC52:C4 | C4xC52:C4 |
| kernel | C4xC52:C4 | C4xC5:D5 | C2xC52:C4 | C52:6C4 | C5xC20 | C52:C4 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C4xC52:C4 ►in GL4(F41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 35 | 35 |
| 0 | 0 | 6 | 40 |
| 40 | 6 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 |
| 28 | 9 | 0 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[0,40,0,0,1,6,0,0,0,0,35,6,0,0,35,40],[40,35,0,0,6,35,0,0,0,0,6,1,0,0,40,0],[0,0,32,28,0,0,0,9,9,0,0,0,0,9,0,0] >;
C4xC52:C4 in GAP, Magma, Sage, TeX
C_4\times C_5^2\rtimes C_4
% in TeX
G:=Group("C4xC5^2:C4"); // GroupNames label
G:=SmallGroup(400,158);
// by ID
G=gap.SmallGroup(400,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1444,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations