metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊5Dic5, Q16⋊5Dic5, SD16⋊3Dic5, (C5×D8)⋊9C4, C5⋊8(C8○D8), (C5×Q16)⋊9C4, C4○D8.5D5, C40.61(C2×C4), (C8×Dic5)⋊2C2, (C5×SD16)⋊8C4, C4.217(D4×D5), C5⋊2C8.35D4, C40.6C4⋊8C2, C4○D4.22D10, C10.129(C4×D4), (C2×C8).254D10, C20.376(C2×D4), D4.Dic5⋊3C2, Q8.3(C2×Dic5), D4.3(C2×Dic5), C8.11(C2×Dic5), C2.16(D4×Dic5), D4⋊2Dic5⋊4C2, (C2×C40).44C22, C4.7(C22×Dic5), C20.136(C22×C4), (C2×C20).467C23, C22.3(D4⋊2D5), C4.Dic5.22C22, (C4×Dic5).277C22, (C5×C4○D8).2C2, (C5×D4).24(C2×C4), (C5×Q8).25(C2×C4), (C5×C4○D4).9C22, (C2×C10).11(C4○D4), (C2×C4).554(C22×D5), (C2×C5⋊2C8).288C22, SmallGroup(320,823)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊5Dic5
G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 278 in 106 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8○D8, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×Dic5, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C8×Dic5, C40.6C4, D4⋊2Dic5, D4.Dic5, C5×C4○D8, D8⋊5Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×Dic5, C22×D5, C8○D8, D4×D5, D4⋊2D5, C22×Dic5, D4×Dic5, D8⋊5Dic5
►On 80 points
(1 30 32 12 36 7 23 17)(2 26 33 13 37 8 24 18)(3 27 34 14 38 9 25 19)(4 28 35 15 39 10 21 20)(5 29 31 11 40 6 22 16)(41 66 77 58 46 61 72 53)(42 67 78 59 47 62 73 54)(43 68 79 60 48 63 74 55)(44 69 80 51 49 64 75 56)(45 70 71 52 50 65 76 57)
(1 46)(2 42)(3 48)(4 44)(5 50)(6 57)(7 53)(8 59)(9 55)(10 51)(11 70)(12 66)(13 62)(14 68)(15 64)(16 65)(17 61)(18 67)(19 63)(20 69)(21 80)(22 76)(23 72)(24 78)(25 74)(26 54)(27 60)(28 56)(29 52)(30 58)(31 71)(32 77)(33 73)(34 79)(35 75)(36 41)(37 47)(38 43)(39 49)(40 45)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 30)(7 29)(8 28)(9 27)(10 26)(11 17)(12 16)(13 20)(14 19)(15 18)(21 33)(22 32)(23 31)(24 35)(25 34)(41 76 46 71)(42 75 47 80)(43 74 48 79)(44 73 49 78)(45 72 50 77)(51 67 56 62)(52 66 57 61)(53 65 58 70)(54 64 59 69)(55 63 60 68)
G:=sub<Sym(80)| (1,30,32,12,36,7,23,17)(2,26,33,13,37,8,24,18)(3,27,34,14,38,9,25,19)(4,28,35,15,39,10,21,20)(5,29,31,11,40,6,22,16)(41,66,77,58,46,61,72,53)(42,67,78,59,47,62,73,54)(43,68,79,60,48,63,74,55)(44,69,80,51,49,64,75,56)(45,70,71,52,50,65,76,57), (1,46)(2,42)(3,48)(4,44)(5,50)(6,57)(7,53)(8,59)(9,55)(10,51)(11,70)(12,66)(13,62)(14,68)(15,64)(16,65)(17,61)(18,67)(19,63)(20,69)(21,80)(22,76)(23,72)(24,78)(25,74)(26,54)(27,60)(28,56)(29,52)(30,58)(31,71)(32,77)(33,73)(34,79)(35,75)(36,41)(37,47)(38,43)(39,49)(40,45), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,30)(7,29)(8,28)(9,27)(10,26)(11,17)(12,16)(13,20)(14,19)(15,18)(21,33)(22,32)(23,31)(24,35)(25,34)(41,76,46,71)(42,75,47,80)(43,74,48,79)(44,73,49,78)(45,72,50,77)(51,67,56,62)(52,66,57,61)(53,65,58,70)(54,64,59,69)(55,63,60,68)>;
G:=Group( (1,30,32,12,36,7,23,17)(2,26,33,13,37,8,24,18)(3,27,34,14,38,9,25,19)(4,28,35,15,39,10,21,20)(5,29,31,11,40,6,22,16)(41,66,77,58,46,61,72,53)(42,67,78,59,47,62,73,54)(43,68,79,60,48,63,74,55)(44,69,80,51,49,64,75,56)(45,70,71,52,50,65,76,57), (1,46)(2,42)(3,48)(4,44)(5,50)(6,57)(7,53)(8,59)(9,55)(10,51)(11,70)(12,66)(13,62)(14,68)(15,64)(16,65)(17,61)(18,67)(19,63)(20,69)(21,80)(22,76)(23,72)(24,78)(25,74)(26,54)(27,60)(28,56)(29,52)(30,58)(31,71)(32,77)(33,73)(34,79)(35,75)(36,41)(37,47)(38,43)(39,49)(40,45), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,30)(7,29)(8,28)(9,27)(10,26)(11,17)(12,16)(13,20)(14,19)(15,18)(21,33)(22,32)(23,31)(24,35)(25,34)(41,76,46,71)(42,75,47,80)(43,74,48,79)(44,73,49,78)(45,72,50,77)(51,67,56,62)(52,66,57,61)(53,65,58,70)(54,64,59,69)(55,63,60,68) );
G=PermutationGroup([[(1,30,32,12,36,7,23,17),(2,26,33,13,37,8,24,18),(3,27,34,14,38,9,25,19),(4,28,35,15,39,10,21,20),(5,29,31,11,40,6,22,16),(41,66,77,58,46,61,72,53),(42,67,78,59,47,62,73,54),(43,68,79,60,48,63,74,55),(44,69,80,51,49,64,75,56),(45,70,71,52,50,65,76,57)], [(1,46),(2,42),(3,48),(4,44),(5,50),(6,57),(7,53),(8,59),(9,55),(10,51),(11,70),(12,66),(13,62),(14,68),(15,64),(16,65),(17,61),(18,67),(19,63),(20,69),(21,80),(22,76),(23,72),(24,78),(25,74),(26,54),(27,60),(28,56),(29,52),(30,58),(31,71),(32,77),(33,73),(34,79),(35,75),(36,41),(37,47),(38,43),(39,49),(40,45)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,30),(7,29),(8,28),(9,27),(10,26),(11,17),(12,16),(13,20),(14,19),(15,18),(21,33),(22,32),(23,31),(24,35),(25,34),(41,76,46,71),(42,75,47,80),(43,74,48,79),(44,73,49,78),(45,72,50,77),(51,67,56,62),(52,66,57,61),(53,65,58,70),(54,64,59,69),(55,63,60,68)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | Dic5 | Dic5 | Dic5 | D10 | C8○D8 | D4×D5 | D4⋊2D5 | D8⋊5Dic5 |
| kernel | D8⋊5Dic5 | C8×Dic5 | C40.6C4 | D4⋊2Dic5 | D4.Dic5 | C5×C4○D8 | C5×D8 | C5×SD16 | C5×Q16 | C5⋊2C8 | C4○D8 | C2×C10 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of D8⋊5Dic5 ►in GL4(𝔽41) generated by
| 14 | 26 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 15 | 0 | 0 |
| 35 | 38 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 9 | 5 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 6 | 40 |
G:=sub<GL(4,GF(41))| [14,0,0,0,26,3,0,0,0,0,1,0,0,0,0,1],[3,35,0,0,15,38,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,40,1,0,0,0,0,0,40,0,0,1,6],[9,0,0,0,5,40,0,0,0,0,1,6,0,0,0,40] >;
D8⋊5Dic5 in GAP, Magma, Sage, TeX
D_8\rtimes_5{\rm Dic}_5 % in TeX
G:=Group("D8:5Dic5"); // GroupNames label
G:=SmallGroup(320,823);
// by ID
G=gap.SmallGroup(320,823);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations