metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.55D4, C23.2Dic5, C10.11M4(2), (C2×C10)⋊4C8, C5⋊4(C22⋊C8), C22⋊(C5⋊2C8), C10.19(C2×C8), (C2×C20).15C4, (C2×C4).94D10, (C2×C4).4Dic5, (C22×C4).1D5, C4.30(C5⋊D4), (C22×C10).9C4, (C22×C20).11C2, C2.3(C4.Dic5), C2.1(C23.D5), C22.9(C2×Dic5), C10.23(C22⋊C4), (C2×C20).108C22, C2.5(C2×C5⋊2C8), (C2×C5⋊2C8)⋊10C2, (C2×C10).47(C2×C4), SmallGroup(160,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.55D4
G = < a,b,c | a20=1, b4=a10, c2=a5, bab-1=cac-1=a9, cbc-1=a15b3 >
Smallest permutation representation of C20.55D4
►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 38 70 42 11 28 80 52)(2 27 71 51 12 37 61 41)(3 36 72 60 13 26 62 50)(4 25 73 49 14 35 63 59)(5 34 74 58 15 24 64 48)(6 23 75 47 16 33 65 57)(7 32 76 56 17 22 66 46)(8 21 77 45 18 31 67 55)(9 30 78 54 19 40 68 44)(10 39 79 43 20 29 69 53)
(1 57 6 42 11 47 16 52)(2 46 7 51 12 56 17 41)(3 55 8 60 13 45 18 50)(4 44 9 49 14 54 19 59)(5 53 10 58 15 43 20 48)(21 77 26 62 31 67 36 72)(22 66 27 71 32 76 37 61)(23 75 28 80 33 65 38 70)(24 64 29 69 34 74 39 79)(25 73 30 78 35 63 40 68)
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), (1,38,70,42,11,28,80,52)(2,27,71,51,12,37,61,41)(3,36,72,60,13,26,62,50)(4,25,73,49,14,35,63,59)(5,34,74,58,15,24,64,48)(6,23,75,47,16,33,65,57)(7,32,76,56,17,22,66,46)(8,21,77,45,18,31,67,55)(9,30,78,54,19,40,68,44)(10,39,79,43,20,29,69,53), (1,57,6,42,11,47,16,52)(2,46,7,51,12,56,17,41)(3,55,8,60,13,45,18,50)(4,44,9,49,14,54,19,59)(5,53,10,58,15,43,20,48)(21,77,26,62,31,67,36,72)(22,66,27,71,32,76,37,61)(23,75,28,80,33,65,38,70)(24,64,29,69,34,74,39,79)(25,73,30,78,35,63,40,68)>;
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), (1,38,70,42,11,28,80,52)(2,27,71,51,12,37,61,41)(3,36,72,60,13,26,62,50)(4,25,73,49,14,35,63,59)(5,34,74,58,15,24,64,48)(6,23,75,47,16,33,65,57)(7,32,76,56,17,22,66,46)(8,21,77,45,18,31,67,55)(9,30,78,54,19,40,68,44)(10,39,79,43,20,29,69,53), (1,57,6,42,11,47,16,52)(2,46,7,51,12,56,17,41)(3,55,8,60,13,45,18,50)(4,44,9,49,14,54,19,59)(5,53,10,58,15,43,20,48)(21,77,26,62,31,67,36,72)(22,66,27,71,32,76,37,61)(23,75,28,80,33,65,38,70)(24,64,29,69,34,74,39,79)(25,73,30,78,35,63,40,68) );
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)], [(1,38,70,42,11,28,80,52),(2,27,71,51,12,37,61,41),(3,36,72,60,13,26,62,50),(4,25,73,49,14,35,63,59),(5,34,74,58,15,24,64,48),(6,23,75,47,16,33,65,57),(7,32,76,56,17,22,66,46),(8,21,77,45,18,31,67,55),(9,30,78,54,19,40,68,44),(10,39,79,43,20,29,69,53)], [(1,57,6,42,11,47,16,52),(2,46,7,51,12,56,17,41),(3,55,8,60,13,45,18,50),(4,44,9,49,14,54,19,59),(5,53,10,58,15,43,20,48),(21,77,26,62,31,67,36,72),(22,66,27,71,32,76,37,61),(23,75,28,80,33,65,38,70),(24,64,29,69,34,74,39,79),(25,73,30,78,35,63,40,68)]])
C20.55D4 is a maximal subgroup of
(C2×D20)⋊C4 C4⋊Dic5⋊C4 C5⋊3(C23⋊C8) (C2×Dic5)⋊C8 C24.Dic5 (C2×C20)⋊C8 C4⋊C4⋊Dic5 C10.29C4≀C2 Dic5.14M4(2) Dic5.9M4(2) C40⋊8C4⋊C2 D5×C22⋊C8 D10⋊7M4(2) C22⋊C8⋊D5 C42.6Dic5 C42.7Dic5 (C2×C10).40D8 C4⋊C4.228D10 C4⋊C4.230D10 C4⋊C4.231D10 C4⋊C4.233D10 C42.187D10 C4⋊C4.236D10 D4×C5⋊2C8 C42.47D10 C20⋊7M4(2) (C2×C10).D8 C4⋊D4.D5 (C2×D4).D10 D20⋊16D4 D20⋊17D4 Dic10⋊17D4 C22⋊Q8.D5 (C2×C10).Q16 C10.(C4○D8) D20.36D4 D20.37D4 Dic10.37D4 C8×C5⋊D4 C40⋊32D4 C40⋊D4 C40⋊18D4 C24.4Dic5 (C2×C10)⋊8D8 (C5×D4).31D4 (C5×Q8)⋊13D4 (C2×C10)⋊8Q16 (D4×C10).24C4 (C5×D4)⋊14D4 (C5×D4).32D4 C60.94D4 C60.212D4 C20.S4
C20.55D4 is a maximal quotient of
(C2×C20)⋊8C8 C24.Dic5 (C2×C20)⋊C8 C20.57D8 C20.26Q16 C40.91D4 C40.D4 C40.92D4 C60.94D4 C60.212D4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C5⋊D4 | C5⋊2C8 | C4.Dic5 |
| kernel | C20.55D4 | C2×C5⋊2C8 | C22×C20 | C2×C20 | C22×C10 | C2×C10 | C20 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 |
Matrix representation of C20.55D4 ►in GL3(𝔽41) generated by
| 9 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 16 |
| 3 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 38 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(41))| [9,0,0,0,18,0,0,0,16],[3,0,0,0,0,40,0,1,0],[38,0,0,0,0,1,0,1,0] >;
C20.55D4 in GAP, Magma, Sage, TeX
C_{20}._{55}D_4 % in TeX
G:=Group("C20.55D4"); // GroupNames label
G:=SmallGroup(160,37);
// by ID
G=gap.SmallGroup(160,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;
// generators/relations
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