metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.10D4, (C2×C4).Dic5, (C2×C4).4D10, (C2×Q8).2D5, (C2×C20).12C4, (Q8×C10).2C2, C4.15(C5⋊D4), C5⋊3(C4.10D4), C4.Dic5.4C2, (C2×C20).19C22, C2.7(C23.D5), C22.4(C2×Dic5), C10.28(C22⋊C4), (C2×C10).50(C2×C4), SmallGroup(160,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.10D4
G = < a,b,c | a20=1, b4=a10, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a5b3 >
Smallest permutation representation of C20.10D4
►On 80 points
Generators in S80
(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 61 16 66 11 71 6 76)(2 80 17 65 12 70 7 75)(3 79 18 64 13 69 8 74)(4 78 19 63 14 68 9 73)(5 77 20 62 15 67 10 72)(21 51 36 56 31 41 26 46)(22 50 37 55 32 60 27 45)(23 49 38 54 33 59 28 44)(24 48 39 53 34 58 29 43)(25 47 40 52 35 57 30 42)
(1 56 6 41 11 46 16 51)(2 45 7 50 12 55 17 60)(3 54 8 59 13 44 18 49)(4 43 9 48 14 53 19 58)(5 52 10 57 15 42 20 47)(21 76 26 61 31 66 36 71)(22 65 27 70 32 75 37 80)(23 74 28 79 33 64 38 69)(24 63 29 68 34 73 39 78)(25 72 30 77 35 62 40 67)
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), (1,61,16,66,11,71,6,76)(2,80,17,65,12,70,7,75)(3,79,18,64,13,69,8,74)(4,78,19,63,14,68,9,73)(5,77,20,62,15,67,10,72)(21,51,36,56,31,41,26,46)(22,50,37,55,32,60,27,45)(23,49,38,54,33,59,28,44)(24,48,39,53,34,58,29,43)(25,47,40,52,35,57,30,42), (1,56,6,41,11,46,16,51)(2,45,7,50,12,55,17,60)(3,54,8,59,13,44,18,49)(4,43,9,48,14,53,19,58)(5,52,10,57,15,42,20,47)(21,76,26,61,31,66,36,71)(22,65,27,70,32,75,37,80)(23,74,28,79,33,64,38,69)(24,63,29,68,34,73,39,78)(25,72,30,77,35,62,40,67)>;
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), (1,61,16,66,11,71,6,76)(2,80,17,65,12,70,7,75)(3,79,18,64,13,69,8,74)(4,78,19,63,14,68,9,73)(5,77,20,62,15,67,10,72)(21,51,36,56,31,41,26,46)(22,50,37,55,32,60,27,45)(23,49,38,54,33,59,28,44)(24,48,39,53,34,58,29,43)(25,47,40,52,35,57,30,42), (1,56,6,41,11,46,16,51)(2,45,7,50,12,55,17,60)(3,54,8,59,13,44,18,49)(4,43,9,48,14,53,19,58)(5,52,10,57,15,42,20,47)(21,76,26,61,31,66,36,71)(22,65,27,70,32,75,37,80)(23,74,28,79,33,64,38,69)(24,63,29,68,34,73,39,78)(25,72,30,77,35,62,40,67) );
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)], [(1,61,16,66,11,71,6,76),(2,80,17,65,12,70,7,75),(3,79,18,64,13,69,8,74),(4,78,19,63,14,68,9,73),(5,77,20,62,15,67,10,72),(21,51,36,56,31,41,26,46),(22,50,37,55,32,60,27,45),(23,49,38,54,33,59,28,44),(24,48,39,53,34,58,29,43),(25,47,40,52,35,57,30,42)], [(1,56,6,41,11,46,16,51),(2,45,7,50,12,55,17,60),(3,54,8,59,13,44,18,49),(4,43,9,48,14,53,19,58),(5,52,10,57,15,42,20,47),(21,76,26,61,31,66,36,71),(22,65,27,70,32,75,37,80),(23,74,28,79,33,64,38,69),(24,63,29,68,34,73,39,78),(25,72,30,77,35,62,40,67)]])
C20.10D4 is a maximal subgroup of
(C2×C4).D20 (C2×Q8).D10 C42.Dic5 C42.3Dic5 D5×C4.10D4 M4(2).21D10 D20.14D4 D20.15D4 C40.44D4 C40.29D4 M4(2).15D10 M4(2).16D10 (D4×C10).29C4 2- 1+4⋊2D5 2- 1+4.2D5 C60.54D4 C60.10D4
C20.10D4 is a maximal quotient of
(C2×C20)⋊C8 (C2×C20).Q8 C42.8D10 C20.10D8 C60.54D4 C60.10D4
31 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | D5 | Dic5 | D10 | C5⋊D4 | C4.10D4 | C20.10D4 |
| kernel | C20.10D4 | C4.Dic5 | Q8×C10 | C2×C20 | C20 | C2×Q8 | C2×C4 | C2×C4 | C4 | C5 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 1 | 4 |
Matrix representation of C20.10D4 ►in GL4(𝔽41) generated by
| 0 | 37 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 10 | 10 | 31 | 8 |
| 28 | 38 | 16 | 10 |
| 26 | 26 | 0 | 29 |
| 15 | 15 | 11 | 12 |
| 15 | 26 | 0 | 0 |
| 0 | 24 | 17 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 1 | 40 | 9 |
| 0 | 1 | 0 | 0 |
| 18 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [0,4,10,28,37,0,10,38,0,0,31,16,0,0,8,10],[26,15,15,0,26,15,26,24,0,11,0,17,29,12,0,0],[0,1,0,18,0,1,1,0,1,40,0,0,0,9,0,40] >;
C20.10D4 in GAP, Magma, Sage, TeX
C_{20}._{10}D_4 % in TeX
G:=Group("C20.10D4"); // GroupNames label
G:=SmallGroup(160,43);
// by ID
G=gap.SmallGroup(160,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,86,579,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^5*b^3>;
// generators/relations
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