direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×2+ 1+4, C20.51C23, C10.19C24, C4○D4⋊3C10, D4⋊4(C2×C10), (C2×D4)⋊6C10, Q8⋊4(C2×C10), (D4×C10)⋊15C2, (C2×C20)⋊9C22, C23⋊2(C2×C10), (C5×D4)⋊13C22, (C2×C10).7C23, C4.9(C22×C10), C2.4(C23×C10), (C5×Q8)⋊12C22, (C22×C10)⋊2C22, C22.2(C22×C10), (C5×D4)○(C5×D4), (C5×Q8)○(C5×Q8), (C2×C4)⋊2(C2×C10), (C5×C4○D4)⋊8C2, SmallGroup(160,232)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×2+ 1+4
G = < a,b,c,d,e | a5=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 220 in 166 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C10, C2×D4, C4○D4, C20, C2×C10, C2×C10, 2+ 1+4, C2×C20, C5×D4, C5×Q8, C22×C10, D4×C10, C5×C4○D4, C5×2+ 1+4
Quotients: C1, C2, C22, C5, C23, C10, C24, C2×C10, 2+ 1+4, C22×C10, C23×C10, C5×2+ 1+4
►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 33 13 27)(2 34 14 28)(3 35 15 29)(4 31 11 30)(5 32 12 26)(6 20 40 21)(7 16 36 22)(8 17 37 23)(9 18 38 24)(10 19 39 25)
(1 17)(2 18)(3 19)(4 20)(5 16)(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 27 13 33)(2 28 14 34)(3 29 15 35)(4 30 11 31)(5 26 12 32)(6 20 40 21)(7 16 36 22)(8 17 37 23)(9 18 38 24)(10 19 39 25)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 30)(7 26)(8 27)(9 28)(10 29)(11 21)(12 22)(13 23)(14 24)(15 25)(31 40)(32 36)(33 37)(34 38)(35 39)
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), (1,33,13,27)(2,34,14,28)(3,35,15,29)(4,31,11,30)(5,32,12,26)(6,20,40,21)(7,16,36,22)(8,17,37,23)(9,18,38,24)(10,19,39,25), (1,17)(2,18)(3,19)(4,20)(5,16)(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), (1,27,13,33)(2,28,14,34)(3,29,15,35)(4,30,11,31)(5,26,12,32)(6,20,40,21)(7,16,36,22)(8,17,37,23)(9,18,38,24)(10,19,39,25), (1,17)(2,18)(3,19)(4,20)(5,16)(6,30)(7,26)(8,27)(9,28)(10,29)(11,21)(12,22)(13,23)(14,24)(15,25)(31,40)(32,36)(33,37)(34,38)(35,39)>;
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), (1,33,13,27)(2,34,14,28)(3,35,15,29)(4,31,11,30)(5,32,12,26)(6,20,40,21)(7,16,36,22)(8,17,37,23)(9,18,38,24)(10,19,39,25), (1,17)(2,18)(3,19)(4,20)(5,16)(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), (1,27,13,33)(2,28,14,34)(3,29,15,35)(4,30,11,31)(5,26,12,32)(6,20,40,21)(7,16,36,22)(8,17,37,23)(9,18,38,24)(10,19,39,25), (1,17)(2,18)(3,19)(4,20)(5,16)(6,30)(7,26)(8,27)(9,28)(10,29)(11,21)(12,22)(13,23)(14,24)(15,25)(31,40)(32,36)(33,37)(34,38)(35,39) );
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)], [(1,33,13,27),(2,34,14,28),(3,35,15,29),(4,31,11,30),(5,32,12,26),(6,20,40,21),(7,16,36,22),(8,17,37,23),(9,18,38,24),(10,19,39,25)], [(1,17),(2,18),(3,19),(4,20),(5,16),(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)], [(1,27,13,33),(2,28,14,34),(3,29,15,35),(4,30,11,31),(5,26,12,32),(6,20,40,21),(7,16,36,22),(8,17,37,23),(9,18,38,24),(10,19,39,25)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,30),(7,26),(8,27),(9,28),(10,29),(11,21),(12,22),(13,23),(14,24),(15,25),(31,40),(32,36),(33,37),(34,38),(35,39)]])
C5×2+ 1+4 is a maximal subgroup of
2+ 1+4⋊D5 2+ 1+4.D5 2+ 1+4.2D5 2+ 1+4⋊2D5 D20.32C23 D20.33C23 D20.37C23
C5×2+ 1+4 is a maximal quotient of C5×D42 C5×Q82
85 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2J | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10AN | 20A | ··· | 20X |
| order | 1 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×2+ 1+4 | D4×C10 | C5×C4○D4 | 2+ 1+4 | C2×D4 | C4○D4 | C5 | C1 |
| # reps | 1 | 9 | 6 | 4 | 36 | 24 | 1 | 4 |
Matrix representation of C5×2+ 1+4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 0 | 1 | 0 |
| 40 | 1 | 40 | 2 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 1 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 40 | 1 | 39 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 40 | 1 | 40 | 2 |
| 1 | 0 | 0 | 0 |
| 1 | 40 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 40 | 1 | 40 | 2 |
| 40 | 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],[0,40,40,0,0,1,0,40,1,40,0,1,0,2,0,40],[0,1,1,0,1,0,40,0,0,0,1,0,0,0,39,40],[0,40,1,1,0,1,0,40,40,40,0,0,0,2,0,40],[0,1,40,40,1,0,1,1,0,0,40,0,0,0,2,1] >;
C5×2+ 1+4 in GAP, Magma, Sage, TeX
C_5\times 2_+^{1+4} % in TeX
G:=Group("C5xES+(2,2)"); // GroupNames label
G:=SmallGroup(160,232);
// by ID
G=gap.SmallGroup(160,232);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,764,2115]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^2=b^2,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=b^2*d>;
// generators/relations