direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5xC4wrC2, D4:2C20, Q8:2C20, C42:3C10, C20.66D4, M4(2):4C10, (C5xD4):8C4, (C5xQ8):8C4, (C4xC20):10C2, C4.3(C2xC20), C4.17(C5xD4), C20.51(C2xC4), C4oD4.1C10, (C2xC10).22D4, C22.3(C5xD4), (C5xM4(2)):10C2, C10.37(C22:C4), (C2xC20).116C22, (C5xC4oD4).4C2, C2.8(C5xC22:C4), (C2xC4).19(C2xC10), SmallGroup(160,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5xC4wrC2
G = < a,b,c,d | a5=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Quotients: C1, C2, C4, C22, C5, C2xC4, D4, C10, C22:C4, C20, C2xC10, C4wrC2, C2xC20, C5xD4, C5xC22:C4, C5xC4wrC2
Smallest permutation representation of C5xC4wrC2
►On 40 points
(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 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)(26 31)(27 32)(28 33)(29 34)(30 35)
G:=sub<Sym(40)| (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), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (1,21)(2,22)(3,23)(4,24)(5,25)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15)(26,31)(27,32)(28,33)(29,34)(30,35)>;
G:=Group( (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), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (1,21)(2,22)(3,23)(4,24)(5,25)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15)(26,31)(27,32)(28,33)(29,34)(30,35) );
G=PermutationGroup([[(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)], [(1,31,21,26),(2,32,22,27),(3,33,23,28),(4,34,24,29),(5,35,25,30),(6,16,36,11),(7,17,37,12),(8,18,38,13),(9,19,39,14),(10,20,40,15)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,16,36,11),(7,17,37,12),(8,18,38,13),(9,19,39,14),(10,20,40,15),(26,31),(27,32),(28,33),(29,34),(30,35)]])
C5xC4wrC2 is a maximal subgroup of
C42:D10 D4:4D20 M4(2).22D10 C42.196D10 M4(2):D10 D4.9D20 D4.10D20
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D4 | D4 | C4wrC2 | C5xD4 | C5xD4 | C5xC4wrC2 |
| kernel | C5xC4wrC2 | C4xC20 | C5xM4(2) | C5xC4oD4 | C5xD4 | C5xQ8 | C4wrC2 | C42 | M4(2) | C4oD4 | D4 | Q8 | C20 | C2xC10 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C5xC4wrC2 ►in GL2(F41) generated by
| 37 | 0 |
| 0 | 37 |
| 9 | 0 |
| 0 | 32 |
| 0 | 32 |
| 9 | 0 |
| 40 | 0 |
| 0 | 32 |
G:=sub<GL(2,GF(41))| [37,0,0,37],[9,0,0,32],[0,9,32,0],[40,0,0,32] >;
C5xC4wrC2 in GAP, Magma, Sage, TeX
C_5\times C_4\wr C_2
% in TeX
G:=Group("C5xC4wrC2"); // GroupNames label
G:=SmallGroup(160,54);
// by ID
G=gap.SmallGroup(160,54);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1209,117,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations
Export