metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.Dic5, Q8.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 D4.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, C4, C4, C22, C5, C8, C2×C4, D4, Q8, C10, C10, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C8○D4, C5⋊2C8, C5⋊2C8, C2×C20, C5×D4, C5×Q8, C2×C5⋊2C8, C4.Dic5, C5×C4○D4, D4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, D10, C8○D4, C2×Dic5, C22×D5, C22×Dic5, D4.Dic5
►On 80 points
(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 41 6 46 11 51 16 56)(2 50 7 55 12 60 17 45)(3 59 8 44 13 49 18 54)(4 48 9 53 14 58 19 43)(5 57 10 42 15 47 20 52)(21 63 26 68 31 73 36 78)(22 72 27 77 32 62 37 67)(23 61 28 66 33 71 38 76)(24 70 29 75 34 80 39 65)(25 79 30 64 35 69 40 74)
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), (1,41,6,46,11,51,16,56)(2,50,7,55,12,60,17,45)(3,59,8,44,13,49,18,54)(4,48,9,53,14,58,19,43)(5,57,10,42,15,47,20,52)(21,63,26,68,31,73,36,78)(22,72,27,77,32,62,37,67)(23,61,28,66,33,71,38,76)(24,70,29,75,34,80,39,65)(25,79,30,64,35,69,40,74)>;
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), (1,41,6,46,11,51,16,56)(2,50,7,55,12,60,17,45)(3,59,8,44,13,49,18,54)(4,48,9,53,14,58,19,43)(5,57,10,42,15,47,20,52)(21,63,26,68,31,73,36,78)(22,72,27,77,32,62,37,67)(23,61,28,66,33,71,38,76)(24,70,29,75,34,80,39,65)(25,79,30,64,35,69,40,74) );
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)], [(1,41,6,46,11,51,16,56),(2,50,7,55,12,60,17,45),(3,59,8,44,13,49,18,54),(4,48,9,53,14,58,19,43),(5,57,10,42,15,47,20,52),(21,63,26,68,31,73,36,78),(22,72,27,77,32,62,37,67),(23,61,28,66,33,71,38,76),(24,70,29,75,34,80,39,65),(25,79,30,64,35,69,40,74)]])
D4.Dic5 is a maximal subgroup of
D4.(C5⋊C8) M4(2).22D10 C42.196D10 D8⋊5Dic5 D8⋊4Dic5 M4(2).D10 M4(2).13D10 M4(2).15D10 M4(2).16D10 C5⋊C16.C22 D5×C8○D4 C20.72C24 C20.76C24 D20.32C23 D20.33C23 D20.34C23 D20.35C23 SL2(𝔽3).Dic5 D12.2Dic5 D12.Dic5 D4.Dic15
D4.Dic5 is a maximal quotient of
C20.35C42 C42.43D10 C42.187D10 D4×C5⋊2C8 C42.47D10 C20⋊7M4(2) Q8×C5⋊2C8 C42.210D10 (D4×C10).24C4 D12.2Dic5 D12.Dic5 D4.Dic15
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 | D4.Dic5 |
| kernel | D4.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 D4.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] >;
D4.Dic5 in GAP, Magma, Sage, TeX
D_4.{\rm Dic}_5 % in TeX
G:=Group("D4.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