?
G = D10⋊10M4(2) order 320 = 26·5
4th semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊10M4(2), D10⋊C8⋊6C2, (C4×D5).112D4, (C2×C10)⋊3M4(2), C22⋊2(C4.F5), (C22×C4).21F5, C23.43(C2×F5), (C22×C20).23C4, C5⋊2(C24.4C4), (C23×D5).16C4, C4.31(C22⋊F5), C23.2F5⋊7C2, C20.30(C22⋊C4), Dic5.106(C2×D4), C10.22(C2×M4(2)), D10.39(C22⋊C4), C22.85(C22×F5), C2.16(D5⋊M4(2)), (C2×Dic5).347C23, (C22×Dic5).275C22, (C2×C5⋊C8)⋊2C22, (C2×C4×D5).37C4, (C2×C4.F5)⋊13C2, C2.11(C2×C4.F5), C2.9(C2×C22⋊F5), C10.6(C2×C22⋊C4), (C2×C4).143(C2×F5), (D5×C22×C4).28C2, (C2×C20).109(C2×C4), (C2×C4×D5).398C22, (C2×C10).63(C22×C4), (C22×C10).63(C2×C4), (C2×Dic5).185(C2×C4), (C22×D5).126(C2×C4), SmallGroup(320,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 762 in 190 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×9], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.4C4, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊C8 [×2], C23.2F5 [×2], C2×C4.F5 [×2], D5×C22×C4, D10⋊10M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×M4(2) [×2], C2×F5 [×3], C24.4C4, C4.F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), C2×C22⋊F5, D10⋊10M4(2)
Generators and relations
G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=ab, dbd=a8b, dcd=c5 >
►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 5)(2 4)(6 10)(7 9)(11 16)(12 15)(13 14)(17 20)(18 19)(21 23)(24 30)(25 29)(26 28)(31 37)(32 36)(33 35)(38 40)(41 45)(42 44)(46 50)(47 49)(51 56)(52 55)(53 54)(57 60)(58 59)(61 68)(62 67)(63 66)(64 65)(69 70)(71 80)(72 79)(73 78)(74 77)(75 76)
(1 14 41 59 30 71 32 65)(2 17 50 56 21 74 31 62)(3 20 49 53 22 77 40 69)(4 13 48 60 23 80 39 66)(5 16 47 57 24 73 38 63)(6 19 46 54 25 76 37 70)(7 12 45 51 26 79 36 67)(8 15 44 58 27 72 35 64)(9 18 43 55 28 75 34 61)(10 11 42 52 29 78 33 68)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 80)(21 24)(22 23)(25 30)(26 29)(27 28)(31 38)(32 37)(33 36)(34 35)(39 40)(41 46)(42 45)(43 44)(47 50)(48 49)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)
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), (1,5)(2,4)(6,10)(7,9)(11,16)(12,15)(13,14)(17,20)(18,19)(21,23)(24,30)(25,29)(26,28)(31,37)(32,36)(33,35)(38,40)(41,45)(42,44)(46,50)(47,49)(51,56)(52,55)(53,54)(57,60)(58,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,80)(72,79)(73,78)(74,77)(75,76), (1,14,41,59,30,71,32,65)(2,17,50,56,21,74,31,62)(3,20,49,53,22,77,40,69)(4,13,48,60,23,80,39,66)(5,16,47,57,24,73,38,63)(6,19,46,54,25,76,37,70)(7,12,45,51,26,79,36,67)(8,15,44,58,27,72,35,64)(9,18,43,55,28,75,34,61)(10,11,42,52,29,78,33,68), (1,6)(2,5)(3,4)(7,10)(8,9)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,80)(21,24)(22,23)(25,30)(26,29)(27,28)(31,38)(32,37)(33,36)(34,35)(39,40)(41,46)(42,45)(43,44)(47,50)(48,49)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69)>;
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), (1,5)(2,4)(6,10)(7,9)(11,16)(12,15)(13,14)(17,20)(18,19)(21,23)(24,30)(25,29)(26,28)(31,37)(32,36)(33,35)(38,40)(41,45)(42,44)(46,50)(47,49)(51,56)(52,55)(53,54)(57,60)(58,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,80)(72,79)(73,78)(74,77)(75,76), (1,14,41,59,30,71,32,65)(2,17,50,56,21,74,31,62)(3,20,49,53,22,77,40,69)(4,13,48,60,23,80,39,66)(5,16,47,57,24,73,38,63)(6,19,46,54,25,76,37,70)(7,12,45,51,26,79,36,67)(8,15,44,58,27,72,35,64)(9,18,43,55,28,75,34,61)(10,11,42,52,29,78,33,68), (1,6)(2,5)(3,4)(7,10)(8,9)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,80)(21,24)(22,23)(25,30)(26,29)(27,28)(31,38)(32,37)(33,36)(34,35)(39,40)(41,46)(42,45)(43,44)(47,50)(48,49)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69) );
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)], [(1,5),(2,4),(6,10),(7,9),(11,16),(12,15),(13,14),(17,20),(18,19),(21,23),(24,30),(25,29),(26,28),(31,37),(32,36),(33,35),(38,40),(41,45),(42,44),(46,50),(47,49),(51,56),(52,55),(53,54),(57,60),(58,59),(61,68),(62,67),(63,66),(64,65),(69,70),(71,80),(72,79),(73,78),(74,77),(75,76)], [(1,14,41,59,30,71,32,65),(2,17,50,56,21,74,31,62),(3,20,49,53,22,77,40,69),(4,13,48,60,23,80,39,66),(5,16,47,57,24,73,38,63),(6,19,46,54,25,76,37,70),(7,12,45,51,26,79,36,67),(8,15,44,58,27,72,35,64),(9,18,43,55,28,75,34,61),(10,11,42,52,29,78,33,68)], [(1,6),(2,5),(3,4),(7,10),(8,9),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,80),(21,24),(22,23),(25,30),(26,29),(27,28),(31,38),(32,37),(33,36),(34,35),(39,40),(41,46),(42,45),(43,44),(47,50),(48,49),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,70),(60,69)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 28 | 28 |
| 0 | 0 | 6 | 40 | 0 | 32 |
| 0 | 0 | 0 | 0 | 40 | 8 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 37 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 13 |
| 0 | 0 | 0 | 1 | 9 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 32 | 25 | 0 | 0 | 0 | 0 |
| 5 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 24 | 26 |
| 0 | 0 | 0 | 40 | 30 | 19 |
| 0 | 0 | 13 | 23 | 1 | 35 |
| 0 | 0 | 9 | 32 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 26 | 21 |
| 0 | 0 | 35 | 35 | 13 | 17 |
| 0 | 0 | 0 | 0 | 7 | 33 |
| 0 | 0 | 0 | 0 | 6 | 34 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,28,0,40,40,0,0,28,32,8,7],[40,37,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,9,40,40,0,0,13,0,0,1],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,0,13,9,0,0,40,40,23,32,0,0,24,30,1,0,0,0,26,19,35,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,26,13,7,6,0,0,21,17,33,34] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22⋊F5 | C4.F5 | D5⋊M4(2) |
| kernel | D10⋊10M4(2) | D10⋊C8 | C23.2F5 | C2×C4.F5 | D5×C22×C4 | C2×C4×D5 | C22×C20 | C23×D5 | C4×D5 | D10 | C2×C10 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{10}\rtimes_{10}M_{4(2)} % in TeX
G:=Group("D10:10M4(2)"); // GroupNames label
G:=SmallGroup(320,1094);
// by ID
G=gap.SmallGroup(320,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^8*b,d*c*d=c^5>;
// generators/relations