metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5⋊M5(2), D5⋊C16⋊4C2, C5⋊C16⋊2C22, (C4×D5).4C8, C5⋊2(C2×M5(2)), (C2×C8).20F5, C8.33(C2×F5), C40.40(C2×C4), C20.15(C2×C8), (C2×C40).24C4, (C8×D5).12C4, C8.F5⋊5C2, C20.C8⋊7C2, C4.10(D5⋊C8), D10.18(C2×C8), (C22×D5).9C8, C4.47(C22×F5), C10.10(C22×C8), C20.87(C22×C4), C5⋊2C8.37C23, C22.5(D5⋊C8), (C2×Dic5).14C8, Dic5.18(C2×C8), (C8×D5).63C22, (D5×C2×C8).31C2, (C2×C4×D5).39C4, C2.11(C2×D5⋊C8), (C2×C10).12(C2×C8), C5⋊2C8.53(C2×C4), (C4×D5).93(C2×C4), (C2×C4).134(C2×F5), (C2×C20).145(C2×C4), (C2×C5⋊2C8).338C22, SmallGroup(320,1053)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 250 in 90 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C16 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2) [×4], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16 [×4], C8×D5 [×4], C2×C5⋊2C8, C2×C40, C2×C4×D5, D5⋊C16 [×2], C8.F5 [×2], C20.C8 [×2], D5×C2×C8, D5⋊M5(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C2×F5 [×3], C2×M5(2), D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D5⋊M5(2)
Generators and relations
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c9 >
►On 80 points
(1 28 39 74 62)(2 75 29 63 40)(3 64 76 41 30)(4 42 49 31 77)(5 32 43 78 50)(6 79 17 51 44)(7 52 80 45 18)(8 46 53 19 65)(9 20 47 66 54)(10 67 21 55 48)(11 56 68 33 22)(12 34 57 23 69)(13 24 35 70 58)(14 71 25 59 36)(15 60 72 37 26)(16 38 61 27 73)
(1 62)(2 40)(3 30)(4 77)(5 50)(6 44)(7 18)(8 65)(9 54)(10 48)(11 22)(12 69)(13 58)(14 36)(15 26)(16 73)(19 46)(20 66)(23 34)(24 70)(27 38)(28 74)(31 42)(32 78)(33 56)(37 60)(41 64)(45 52)(51 79)(55 67)(59 71)(63 75)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)
G:=sub<Sym(80)| (1,28,39,74,62)(2,75,29,63,40)(3,64,76,41,30)(4,42,49,31,77)(5,32,43,78,50)(6,79,17,51,44)(7,52,80,45,18)(8,46,53,19,65)(9,20,47,66,54)(10,67,21,55,48)(11,56,68,33,22)(12,34,57,23,69)(13,24,35,70,58)(14,71,25,59,36)(15,60,72,37,26)(16,38,61,27,73), (1,62)(2,40)(3,30)(4,77)(5,50)(6,44)(7,18)(8,65)(9,54)(10,48)(11,22)(12,69)(13,58)(14,36)(15,26)(16,73)(19,46)(20,66)(23,34)(24,70)(27,38)(28,74)(31,42)(32,78)(33,56)(37,60)(41,64)(45,52)(51,79)(55,67)(59,71)(63,75), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)>;
G:=Group( (1,28,39,74,62)(2,75,29,63,40)(3,64,76,41,30)(4,42,49,31,77)(5,32,43,78,50)(6,79,17,51,44)(7,52,80,45,18)(8,46,53,19,65)(9,20,47,66,54)(10,67,21,55,48)(11,56,68,33,22)(12,34,57,23,69)(13,24,35,70,58)(14,71,25,59,36)(15,60,72,37,26)(16,38,61,27,73), (1,62)(2,40)(3,30)(4,77)(5,50)(6,44)(7,18)(8,65)(9,54)(10,48)(11,22)(12,69)(13,58)(14,36)(15,26)(16,73)(19,46)(20,66)(23,34)(24,70)(27,38)(28,74)(31,42)(32,78)(33,56)(37,60)(41,64)(45,52)(51,79)(55,67)(59,71)(63,75), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79) );
G=PermutationGroup([(1,28,39,74,62),(2,75,29,63,40),(3,64,76,41,30),(4,42,49,31,77),(5,32,43,78,50),(6,79,17,51,44),(7,52,80,45,18),(8,46,53,19,65),(9,20,47,66,54),(10,67,21,55,48),(11,56,68,33,22),(12,34,57,23,69),(13,24,35,70,58),(14,71,25,59,36),(15,60,72,37,26),(16,38,61,27,73)], [(1,62),(2,40),(3,30),(4,77),(5,50),(6,44),(7,18),(8,65),(9,54),(10,48),(11,22),(12,69),(13,58),(14,36),(15,26),(16,73),(19,46),(20,66),(23,34),(24,70),(27,38),(28,74),(31,42),(32,78),(33,56),(37,60),(41,64),(45,52),(51,79),(55,67),(59,71),(63,75)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 190 | 191 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 0 | 190 |
| 0 | 0 | 52 | 51 |
| 240 | 50 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 84 | 84 | 0 | 0 |
| 30 | 157 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [190,240,0,0,191,240,0,0,0,0,0,52,0,0,190,51],[240,0,0,0,50,1,0,0,0,0,1,240,0,0,0,240],[0,0,84,30,0,0,84,157,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | D5⋊M5(2) |
| kernel | D5⋊M5(2) | D5⋊C16 | C8.F5 | C20.C8 | D5×C2×C8 | C8×D5 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_5\rtimes M_{5(2)} % in TeX
G:=Group("D5:M5(2)"); // GroupNames label
G:=SmallGroup(320,1053);
// by ID
G=gap.SmallGroup(320,1053);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^9>;
// generators/relations