direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×M4(2), C40⋊14C22, C23.3C20, C20.53C23, (C2×C8)⋊6C10, C8⋊4(C2×C10), (C2×C40)⋊14C2, (C2×C4).6C20, C4.10(C2×C20), (C2×C20).26C4, C20.68(C2×C4), C2.6(C22×C20), (C22×C10).8C4, C22.6(C2×C20), (C22×C4).6C10, (C22×C20).16C2, C10.47(C22×C4), C4.11(C22×C10), (C2×C20).127C22, (C2×C10).43(C2×C4), (C2×C4).33(C2×C10), SmallGroup(160,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×M4(2)
G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C2×C40, C5×M4(2), C22×C20, C10×M4(2)
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, M4(2), C22×C4, C20, C2×C10, C2×M4(2), C2×C20, C22×C10, C5×M4(2), C22×C20, C10×M4(2)
►On 80 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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 20 50 38 70 28 52 79)(2 11 41 39 61 29 53 80)(3 12 42 40 62 30 54 71)(4 13 43 31 63 21 55 72)(5 14 44 32 64 22 56 73)(6 15 45 33 65 23 57 74)(7 16 46 34 66 24 58 75)(8 17 47 35 67 25 59 76)(9 18 48 36 68 26 60 77)(10 19 49 37 69 27 51 78)
(11 29)(12 30)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 71)
G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,20,50,38,70,28,52,79)(2,11,41,39,61,29,53,80)(3,12,42,40,62,30,54,71)(4,13,43,31,63,21,55,72)(5,14,44,32,64,22,56,73)(6,15,45,33,65,23,57,74)(7,16,46,34,66,24,58,75)(8,17,47,35,67,25,59,76)(9,18,48,36,68,26,60,77)(10,19,49,37,69,27,51,78), (11,29)(12,30)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,71)>;
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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,20,50,38,70,28,52,79)(2,11,41,39,61,29,53,80)(3,12,42,40,62,30,54,71)(4,13,43,31,63,21,55,72)(5,14,44,32,64,22,56,73)(6,15,45,33,65,23,57,74)(7,16,46,34,66,24,58,75)(8,17,47,35,67,25,59,76)(9,18,48,36,68,26,60,77)(10,19,49,37,69,27,51,78), (11,29)(12,30)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,71) );
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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,20,50,38,70,28,52,79),(2,11,41,39,61,29,53,80),(3,12,42,40,62,30,54,71),(4,13,43,31,63,21,55,72),(5,14,44,32,64,22,56,73),(6,15,45,33,65,23,57,74),(7,16,46,34,66,24,58,75),(8,17,47,35,67,25,59,76),(9,18,48,36,68,26,60,77),(10,19,49,37,69,27,51,78)], [(11,29),(12,30),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,71)]])
C10×M4(2) is a maximal subgroup of
C40.D4 M4(2)⋊Dic5 C20.33C42 (C2×C40)⋊C4 C23.9D20 C20.34C42 M4(2)⋊4Dic5 C20.51C42 Dic5⋊5M4(2) C20.51(C4⋊C4) C23.46D20 C23.47D20 C20.37C42 C23.Dic10 M4(2).Dic5 D10⋊8M4(2) C40⋊D4 C40⋊18D4 C4.89(C2×D20) C23.48D20 M4(2).31D10 C23.49D20 C40⋊2D4 C40⋊3D4 C40.4D4 C23.20D20 C40.47C23 C40.9C23
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | M4(2) | C5×M4(2) |
| kernel | C10×M4(2) | C2×C40 | C5×M4(2) | C22×C20 | C2×C20 | C22×C10 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 8 | 16 | 4 | 24 | 8 | 4 | 16 |
Matrix representation of C10×M4(2) ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 25 |
| 32 | 0 | 0 |
| 0 | 9 | 39 |
| 0 | 4 | 32 |
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 9 | 40 |
G:=sub<GL(3,GF(41))| [40,0,0,0,25,0,0,0,25],[32,0,0,0,9,4,0,39,32],[40,0,0,0,1,9,0,0,40] >;
C10×M4(2) in GAP, Magma, Sage, TeX
C_{10}\times M_4(2) % in TeX
G:=Group("C10xM4(2)"); // GroupNames label
G:=SmallGroup(160,191);
// by ID
G=gap.SmallGroup(160,191);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,985,88]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations