direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C42⋊C4, C42⋊2C20, (C4×C20)⋊15C4, (C2×D4)⋊2C20, (D4×C10)⋊16C4, C23⋊C4⋊2C10, C23.3(C5×D4), C4⋊1D4.2C10, (C22×C10).3D4, C10.55(C23⋊C4), (D4×C10).176C22, (C5×C23⋊C4)⋊8C2, (C2×C4).1(C2×C20), C2.8(C5×C23⋊C4), (C2×D4).3(C2×C10), (C5×C4⋊1D4).9C2, (C2×C20).185(C2×C4), C22.12(C5×C22⋊C4), (C2×C10).139(C22⋊C4), SmallGroup(320,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >
Subgroups: 242 in 86 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C10, C10, C42, C22⋊C4, C2×D4, C2×D4, C20, C2×C10, C2×C10, C23⋊C4, C4⋊1D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C42⋊C4, C4×C20, C5×C22⋊C4, D4×C10, D4×C10, C5×C23⋊C4, C5×C4⋊1D4, C5×C42⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C23⋊C4, C2×C20, C5×D4, C42⋊C4, C5×C22⋊C4, C5×C23⋊C4, C5×C42⋊C4
►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)
(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)
(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 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26 16 31)(12 27 17 32)(13 28 18 33)(14 29 19 34)(15 30 20 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), (6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26,16,31)(12,27,17,32)(13,28,18,33)(14,29,19,34)(15,30,20,35)>;
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), (6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26,16,31)(12,27,17,32)(13,28,18,33)(14,29,19,34)(15,30,20,35) );
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)], [(6,16,36,11),(7,17,37,12),(8,18,38,13),(9,19,39,14),(10,20,40,15)], [(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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26,16,31),(12,27,17,32),(13,28,18,33),(14,29,19,34),(15,30,20,35)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 10Q | 10R | 10S | 10T | 20A | ··· | 20L | 20M | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | C5×D4 | C23⋊C4 | C42⋊C4 | C5×C23⋊C4 | C5×C42⋊C4 |
| kernel | C5×C42⋊C4 | C5×C23⋊C4 | C5×C4⋊1D4 | C4×C20 | D4×C10 | C42⋊C4 | C23⋊C4 | C4⋊1D4 | C42 | C2×D4 | C22×C10 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C42⋊C4 ►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 |
| 1 | 1 | 1 | 2 |
| 40 | 40 | 40 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 1 | 1 | 1 | 2 |
| 0 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 40 | 40 | 39 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,1,40,0,1,1,40,0,0,1,40,0,0,2,40],[0,40,1,0,1,0,1,40,0,0,1,40,0,0,2,40],[0,40,1,0,0,40,0,1,1,40,0,0,0,39,0,1] >;
C5×C42⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes C_4
% in TeX
G:=Group("C5xC4^2:C4"); // GroupNames label
G:=SmallGroup(320,158);
// by ID
G=gap.SmallGroup(320,158);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,375,10085]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
// generators/relations