metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊15D10, Q16⋊13D10, D20.45D4, SD16⋊11D10, D40⋊19C22, C40.38C23, C20.16C24, Dic10.45D4, D20.11C23, C4○D8⋊4D5, (D5×D8)⋊7C2, C5⋊3(D4○D8), C4○D4⋊1D10, (C2×C8)⋊13D10, C5⋊D4.1D4, (C2×D40)⋊23C2, D40⋊C2⋊6C2, D4⋊D5⋊3C22, C4.143(D4×D5), (D4×D5)⋊2C22, (C8×D5)⋊8C22, Q8⋊D5⋊2C22, D4⋊D10⋊7C2, Q8.D10⋊7C2, D4⋊8D10⋊5C2, C22.8(D4×D5), (C2×C40)⋊12C22, D10.52(C2×D4), C20.349(C2×D4), (C5×D8)⋊13C22, C5⋊2C8.7C23, (C4×D5).9C23, C4.16(C23×D5), C8.16(C22×D5), D20.3C4⋊7C2, (C2×D20)⋊34C22, C8⋊D5⋊11C22, Dic5.58(C2×D4), (C5×Q16)⋊11C22, Q8⋊2D5⋊2C22, (C5×D4).10C23, D4.10(C22×D5), Q8.10(C22×D5), (C5×Q8).10C23, (C2×C20).533C23, C4○D20.54C22, (C5×SD16)⋊11C22, C10.117(C22×D4), C4.Dic5⋊30C22, C2.90(C2×D4×D5), (C5×C4○D8)⋊5C2, (C2×C10).13(C2×D4), (C5×C4○D4)⋊3C22, (C2×C4).232(C22×D5), SmallGroup(320,1441)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1238 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×2], D4 [×19], Q8 [×2], Q8, C23 [×6], D5 [×6], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×8], SD16 [×2], SD16 [×4], Q16, C2×D4 [×12], C4○D4 [×2], C4○D4 [×7], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×10], C2×C10, C2×C10 [×2], C8○D4, C2×D8 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×6], 2+ (1+4) [×2], C5⋊2C8 [×2], C40 [×2], Dic10, C4×D5 [×2], C4×D5 [×4], D20, D20 [×4], D20 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×6], D4○D8, C8×D5 [×2], C8⋊D5 [×2], D40 [×4], C4.Dic5, D4⋊D5 [×4], Q8⋊D5 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×D20 [×2], C2×D20 [×2], C4○D20, C4○D20 [×2], D4×D5 [×4], D4×D5 [×4], Q8⋊2D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×D40, D5×D8 [×2], D40⋊C2 [×4], Q8.D10 [×2], D4⋊D10 [×2], C5×C4○D8, D4⋊8D10 [×2], D8⋊15D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D8⋊15D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >
►On 80 points
(1 22 55 46 63 73 40 12)(2 23 56 47 64 74 31 13)(3 24 57 48 65 75 32 14)(4 25 58 49 66 76 33 15)(5 26 59 50 67 77 34 16)(6 27 60 41 68 78 35 17)(7 28 51 42 69 79 36 18)(8 29 52 43 70 80 37 19)(9 30 53 44 61 71 38 20)(10 21 54 45 62 72 39 11)
(2 64)(4 66)(6 68)(8 70)(10 62)(11 72)(12 22)(13 74)(14 24)(15 76)(16 26)(17 78)(18 28)(19 80)(20 30)(21 45)(23 47)(25 49)(27 41)(29 43)(32 57)(34 59)(36 51)(38 53)(40 55)(42 79)(44 71)(46 73)(48 75)(50 77)
(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 21)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 55)(12 54)(13 53)(14 52)(15 51)(16 60)(17 59)(18 58)(19 57)(20 56)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)
G:=sub<Sym(80)| (1,22,55,46,63,73,40,12)(2,23,56,47,64,74,31,13)(3,24,57,48,65,75,32,14)(4,25,58,49,66,76,33,15)(5,26,59,50,67,77,34,16)(6,27,60,41,68,78,35,17)(7,28,51,42,69,79,36,18)(8,29,52,43,70,80,37,19)(9,30,53,44,61,71,38,20)(10,21,54,45,62,72,39,11), (2,64)(4,66)(6,68)(8,70)(10,62)(11,72)(12,22)(13,74)(14,24)(15,76)(16,26)(17,78)(18,28)(19,80)(20,30)(21,45)(23,47)(25,49)(27,41)(29,43)(32,57)(34,59)(36,51)(38,53)(40,55)(42,79)(44,71)(46,73)(48,75)(50,77), (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), (1,21)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,55)(12,54)(13,53)(14,52)(15,51)(16,60)(17,59)(18,58)(19,57)(20,56)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)>;
G:=Group( (1,22,55,46,63,73,40,12)(2,23,56,47,64,74,31,13)(3,24,57,48,65,75,32,14)(4,25,58,49,66,76,33,15)(5,26,59,50,67,77,34,16)(6,27,60,41,68,78,35,17)(7,28,51,42,69,79,36,18)(8,29,52,43,70,80,37,19)(9,30,53,44,61,71,38,20)(10,21,54,45,62,72,39,11), (2,64)(4,66)(6,68)(8,70)(10,62)(11,72)(12,22)(13,74)(14,24)(15,76)(16,26)(17,78)(18,28)(19,80)(20,30)(21,45)(23,47)(25,49)(27,41)(29,43)(32,57)(34,59)(36,51)(38,53)(40,55)(42,79)(44,71)(46,73)(48,75)(50,77), (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), (1,21)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,55)(12,54)(13,53)(14,52)(15,51)(16,60)(17,59)(18,58)(19,57)(20,56)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75) );
G=PermutationGroup([(1,22,55,46,63,73,40,12),(2,23,56,47,64,74,31,13),(3,24,57,48,65,75,32,14),(4,25,58,49,66,76,33,15),(5,26,59,50,67,77,34,16),(6,27,60,41,68,78,35,17),(7,28,51,42,69,79,36,18),(8,29,52,43,70,80,37,19),(9,30,53,44,61,71,38,20),(10,21,54,45,62,72,39,11)], [(2,64),(4,66),(6,68),(8,70),(10,62),(11,72),(12,22),(13,74),(14,24),(15,76),(16,26),(17,78),(18,28),(19,80),(20,30),(21,45),(23,47),(25,49),(27,41),(29,43),(32,57),(34,59),(36,51),(38,53),(40,55),(42,79),(44,71),(46,73),(48,75),(50,77)], [(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)], [(1,21),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,55),(12,54),(13,53),(14,52),(15,51),(16,60),(17,59),(18,58),(19,57),(20,56),(31,44),(32,43),(33,42),(34,41),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(61,74),(62,73),(63,72),(64,71),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 29 | 0 | 29 | 0 |
| 0 | 29 | 0 | 29 |
| 12 | 0 | 29 | 0 |
| 0 | 12 | 0 | 29 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 27 |
| 0 | 0 | 14 | 14 |
| 11 | 14 | 0 | 0 |
| 27 | 27 | 0 | 0 |
| 9 | 15 | 32 | 26 |
| 37 | 32 | 4 | 9 |
| 32 | 26 | 32 | 26 |
| 4 | 9 | 4 | 9 |
G:=sub<GL(4,GF(41))| [29,0,12,0,0,29,0,12,29,0,29,0,0,29,0,29],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,11,27,0,0,14,27,30,14,0,0,27,14,0,0],[9,37,32,4,15,32,26,9,32,4,32,4,26,9,26,9] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D4○D8 | D4×D5 | D4×D5 | D8⋊15D10 |
| kernel | D8⋊15D10 | D20.3C4 | C2×D40 | D5×D8 | D40⋊C2 | Q8.D10 | D4⋊D10 | C5×C4○D8 | D4⋊8D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{15}D_{10} % in TeX
G:=Group("D8:15D10"); // GroupNames label
G:=SmallGroup(320,1441);
// by ID
G=gap.SmallGroup(320,1441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations