metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.Dic5, D4.Dic5, C20.42C23, C5⋊5(C8○D4), C4○D4.3D5, (C5×D4).2C4, (C5×Q8).2C4, C20.36(C2×C4), (C2×C4).58D10, C4.Dic5⋊8C2, C4.5(C2×Dic5), C4.42(C22×D5), (C2×C20).41C22, C10.40(C22×C4), C5⋊2C8.13C22, C2.8(C22×Dic5), C22.1(C2×Dic5), (C2×C5⋊2C8)⋊7C2, (C5×C4○D4).2C2, (C2×C10).27(C2×C4), SmallGroup(160,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.Dic5
G = < a,b,c,d | a4=1, b2=c10=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >
Subgroups: 112 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C8○D4, C5⋊2C8, C5⋊2C8 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×C5⋊2C8 [×3], C4.Dic5 [×3], C5×C4○D4, Q8.Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, Dic5 [×4], D10 [×3], C8○D4, C2×Dic5 [×6], C22×D5, C22×Dic5, Q8.Dic5
►On 80 points
Generators in S80
(1 38 11 28)(2 39 12 29)(3 40 13 30)(4 21 14 31)(5 22 15 32)(6 23 16 33)(7 24 17 34)(8 25 18 35)(9 26 19 36)(10 27 20 37)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 61 56 71)(47 62 57 72)(48 63 58 73)(49 64 59 74)(50 65 60 75)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(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 51 6 56 11 41 16 46)(2 60 7 45 12 50 17 55)(3 49 8 54 13 59 18 44)(4 58 9 43 14 48 19 53)(5 47 10 52 15 57 20 42)(21 73 26 78 31 63 36 68)(22 62 27 67 32 72 37 77)(23 71 28 76 33 61 38 66)(24 80 29 65 34 70 39 75)(25 69 30 74 35 79 40 64)
G:=sub<Sym(80)| (1,38,11,28)(2,39,12,29)(3,40,13,30)(4,21,14,31)(5,22,15,32)(6,23,16,33)(7,24,17,34)(8,25,18,35)(9,26,19,36)(10,27,20,37)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (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), (1,51,6,56,11,41,16,46)(2,60,7,45,12,50,17,55)(3,49,8,54,13,59,18,44)(4,58,9,43,14,48,19,53)(5,47,10,52,15,57,20,42)(21,73,26,78,31,63,36,68)(22,62,27,67,32,72,37,77)(23,71,28,76,33,61,38,66)(24,80,29,65,34,70,39,75)(25,69,30,74,35,79,40,64)>;
G:=Group( (1,38,11,28)(2,39,12,29)(3,40,13,30)(4,21,14,31)(5,22,15,32)(6,23,16,33)(7,24,17,34)(8,25,18,35)(9,26,19,36)(10,27,20,37)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,61,56,71)(47,62,57,72)(48,63,58,73)(49,64,59,74)(50,65,60,75), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (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), (1,51,6,56,11,41,16,46)(2,60,7,45,12,50,17,55)(3,49,8,54,13,59,18,44)(4,58,9,43,14,48,19,53)(5,47,10,52,15,57,20,42)(21,73,26,78,31,63,36,68)(22,62,27,67,32,72,37,77)(23,71,28,76,33,61,38,66)(24,80,29,65,34,70,39,75)(25,69,30,74,35,79,40,64) );
G=PermutationGroup([(1,38,11,28),(2,39,12,29),(3,40,13,30),(4,21,14,31),(5,22,15,32),(6,23,16,33),(7,24,17,34),(8,25,18,35),(9,26,19,36),(10,27,20,37),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,61,56,71),(47,62,57,72),(48,63,58,73),(49,64,59,74),(50,65,60,75)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)], [(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)], [(1,51,6,56,11,41,16,46),(2,60,7,45,12,50,17,55),(3,49,8,54,13,59,18,44),(4,58,9,43,14,48,19,53),(5,47,10,52,15,57,20,42),(21,73,26,78,31,63,36,68),(22,62,27,67,32,72,37,77),(23,71,28,76,33,61,38,66),(24,80,29,65,34,70,39,75),(25,69,30,74,35,79,40,64)])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | D10 | Dic5 | Dic5 | C8○D4 | Q8.Dic5 |
| kernel | Q8.Dic5 | C2×C5⋊2C8 | C4.Dic5 | C5×C4○D4 | C5×D4 | C5×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 6 | 6 | 2 | 4 | 4 |
Matrix representation of Q8.Dic5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 9 | 32 |
| 40 | 1 | 0 | 0 |
| 33 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 35 | 0 | 0 |
| 34 | 0 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,23,32],[1,0,0,0,0,1,0,0,0,0,9,9,0,0,0,32],[40,33,0,0,1,7,0,0,0,0,9,0,0,0,0,9],[0,34,0,0,35,0,0,0,0,0,38,0,0,0,0,38] >;
Q8.Dic5 in GAP, Magma, Sage, TeX
Q_8.{\rm Dic}_5 % in TeX
G:=Group("Q8.Dic5"); // GroupNames label
G:=SmallGroup(160,169);
// by ID
G=gap.SmallGroup(160,169);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^10=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations