direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊C22, D8⋊2C10, C40⋊7C22, C20.63D4, SD16⋊1C10, M4(2)⋊1C10, C20.48C23, C8⋊(C2×C10), (C5×D8)⋊6C2, C4○D4⋊2C10, (C2×D4)⋊5C10, D4⋊2(C2×C10), Q8⋊2(C2×C10), C4.14(C5×D4), (D4×C10)⋊14C2, (C5×SD16)⋊5C2, C2.15(D4×C10), C10.78(C2×D4), (C2×C10).24D4, C22.5(C5×D4), (C5×D4)⋊11C22, (C5×M4(2))⋊5C2, C4.5(C22×C10), (C5×Q8)⋊10C22, (C2×C20).69C22, (C5×C4○D4)⋊7C2, (C2×C4).10(C2×C10), SmallGroup(160,197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊C22
G = < a,b,c,d | a5=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, D4, Q8, C23, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×C10, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C5×C8⋊C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C8⋊C22, C5×D4, C22×C10, D4×C10, C5×C8⋊C22
►On 40 points
(1 18 31 35 14)(2 19 32 36 15)(3 20 25 37 16)(4 21 26 38 9)(5 22 27 39 10)(6 23 28 40 11)(7 24 29 33 12)(8 17 30 34 13)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)
G:=sub<Sym(40)| (1,18,31,35,14)(2,19,32,36,15)(3,20,25,37,16)(4,21,26,38,9)(5,22,27,39,10)(6,23,28,40,11)(7,24,29,33,12)(8,17,30,34,13), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)>;
G:=Group( (1,18,31,35,14)(2,19,32,36,15)(3,20,25,37,16)(4,21,26,38,9)(5,22,27,39,10)(6,23,28,40,11)(7,24,29,33,12)(8,17,30,34,13), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40) );
G=PermutationGroup([[(1,18,31,35,14),(2,19,32,36,15),(3,20,25,37,16),(4,21,26,38,9),(5,22,27,39,10),(6,23,28,40,11),(7,24,29,33,12),(8,17,30,34,13)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40)]])
C5×C8⋊C22 is a maximal subgroup of
D20⋊18D4 M4(2).D10 M4(2).13D10 D20.38D4 SD16⋊D10 D8⋊5D10 D8⋊6D10
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10T | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C8⋊C22 | C5×M4(2) | C5×D8 | C5×SD16 | D4×C10 | C5×C4○D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C20 | C2×C10 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 | 1 | 4 |
Matrix representation of C5×C8⋊C22 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,0,1,0,0,1,0,1,0,0,0,0,40,0,0],[1,0,0,0,0,40,0,0,0,0,0,40,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40] >;
C5×C8⋊C22 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes C_2^2
% in TeX
G:=Group("C5xC8:C2^2"); // GroupNames label
G:=SmallGroup(160,197);
// by ID
G=gap.SmallGroup(160,197);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations