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 [×4], C4 [×5], C22, C22 [×7], C5, C2×C4, C2×C4 [×3], D4 [×6], C23 [×2], C23, C10, C10 [×4], C42, C22⋊C4 [×2], C2×D4 [×2], C2×D4 [×2], C20 [×5], C2×C10, C2×C10 [×7], C23⋊C4 [×2], C4⋊1D4, C2×C20, C2×C20 [×3], C5×D4 [×6], C22×C10 [×2], C22×C10, C42⋊C4, C4×C20, C5×C22⋊C4 [×2], D4×C10 [×2], D4×C10 [×2], C5×C23⋊C4 [×2], C5×C4⋊1D4, C5×C42⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C23⋊C4, C2×C20, C5×D4 [×2], 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 14 20 40)(7 15 16 36)(8 11 17 37)(9 12 18 38)(10 13 19 39)
(1 27 21 35)(2 28 22 31)(3 29 23 32)(4 30 24 33)(5 26 25 34)(6 14 20 40)(7 15 16 36)(8 11 17 37)(9 12 18 38)(10 13 19 39)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 34 20 26)(7 35 16 27)(8 31 17 28)(9 32 18 29)(10 33 19 30)(11 22)(12 23)(13 24)(14 25)(15 21)
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), (6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39), (1,27,21,35)(2,28,22,31)(3,29,23,32)(4,30,24,33)(5,26,25,34)(6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39), (1,36)(2,37)(3,38)(4,39)(5,40)(6,34,20,26)(7,35,16,27)(8,31,17,28)(9,32,18,29)(10,33,19,30)(11,22)(12,23)(13,24)(14,25)(15,21)>;
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), (6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39), (1,27,21,35)(2,28,22,31)(3,29,23,32)(4,30,24,33)(5,26,25,34)(6,14,20,40)(7,15,16,36)(8,11,17,37)(9,12,18,38)(10,13,19,39), (1,36)(2,37)(3,38)(4,39)(5,40)(6,34,20,26)(7,35,16,27)(8,31,17,28)(9,32,18,29)(10,33,19,30)(11,22)(12,23)(13,24)(14,25)(15,21) );
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)], [(6,14,20,40),(7,15,16,36),(8,11,17,37),(9,12,18,38),(10,13,19,39)], [(1,27,21,35),(2,28,22,31),(3,29,23,32),(4,30,24,33),(5,26,25,34),(6,14,20,40),(7,15,16,36),(8,11,17,37),(9,12,18,38),(10,13,19,39)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,34,20,26),(7,35,16,27),(8,31,17,28),(9,32,18,29),(10,33,19,30),(11,22),(12,23),(13,24),(14,25),(15,21)])
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