metabelian, supersoluble, monomial, A-group
Aliases: C20.14F5, C20.3Dic5, D10.2Dic5, Dic5.12D10, D5⋊(C5⋊2C8), (C5×D5)⋊5C8, C52⋊6(C2×C8), C5⋊5(D5⋊C8), (C4×D5).5D5, (C5×C20).10C4, C52⋊3C8⋊7C2, C10.35(C2×F5), (D5×C20).12C2, (D5×C10).10C4, C4.3(D5.D5), C10.1(C2×Dic5), (C5×Dic5).16C22, C5⋊1(C2×C5⋊2C8), C2.1(C2×D5.D5), (C5×C10).20(C2×C4), SmallGroup(400,142)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C20.14F5 |
Generators and relations for C20.14F5
G = < a,b,c | a20=b5=1, c4=a10, ab=ba, cac-1=a9, cbc-1=b3 >
Smallest permutation representation of C20.14F5
►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 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)(41 57 53 49 45)(42 58 54 50 46)(43 59 55 51 47)(44 60 56 52 48)(61 69 77 65 73)(62 70 78 66 74)(63 71 79 67 75)(64 72 80 68 76)
(1 66 43 25 11 76 53 35)(2 75 44 34 12 65 54 24)(3 64 45 23 13 74 55 33)(4 73 46 32 14 63 56 22)(5 62 47 21 15 72 57 31)(6 71 48 30 16 61 58 40)(7 80 49 39 17 70 59 29)(8 69 50 28 18 79 60 38)(9 78 51 37 19 68 41 27)(10 67 52 26 20 77 42 36)
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), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32)(41,57,53,49,45)(42,58,54,50,46)(43,59,55,51,47)(44,60,56,52,48)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,66,43,25,11,76,53,35)(2,75,44,34,12,65,54,24)(3,64,45,23,13,74,55,33)(4,73,46,32,14,63,56,22)(5,62,47,21,15,72,57,31)(6,71,48,30,16,61,58,40)(7,80,49,39,17,70,59,29)(8,69,50,28,18,79,60,38)(9,78,51,37,19,68,41,27)(10,67,52,26,20,77,42,36)>;
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), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32)(41,57,53,49,45)(42,58,54,50,46)(43,59,55,51,47)(44,60,56,52,48)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,66,43,25,11,76,53,35)(2,75,44,34,12,65,54,24)(3,64,45,23,13,74,55,33)(4,73,46,32,14,63,56,22)(5,62,47,21,15,72,57,31)(6,71,48,30,16,61,58,40)(7,80,49,39,17,70,59,29)(8,69,50,28,18,79,60,38)(9,78,51,37,19,68,41,27)(10,67,52,26,20,77,42,36) );
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)], [(1,5,9,13,17),(2,6,10,14,18),(3,7,11,15,19),(4,8,12,16,20),(21,33,25,37,29),(22,34,26,38,30),(23,35,27,39,31),(24,36,28,40,32),(41,57,53,49,45),(42,58,54,50,46),(43,59,55,51,47),(44,60,56,52,48),(61,69,77,65,73),(62,70,78,66,74),(63,71,79,67,75),(64,72,80,68,76)], [(1,66,43,25,11,76,53,35),(2,75,44,34,12,65,54,24),(3,64,45,23,13,74,55,33),(4,73,46,32,14,63,56,22),(5,62,47,21,15,72,57,31),(6,71,48,30,16,61,58,40),(7,80,49,39,17,70,59,29),(8,69,50,28,18,79,60,38),(9,78,51,37,19,68,41,27),(10,67,52,26,20,77,42,36)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | ··· | 5G | 8A | ··· | 8H | 10A | 10B | 10C | ··· | 10G | 10H | 10I | 10J | 10K | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 4 | ··· | 4 | 25 | ··· | 25 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | D10 | Dic5 | Dic5 | C5⋊2C8 | F5 | C2×F5 | D5⋊C8 | D5.D5 | C2×D5.D5 | C20.14F5 |
| kernel | C20.14F5 | C52⋊3C8 | D5×C20 | C5×C20 | D5×C10 | C5×D5 | C4×D5 | Dic5 | C20 | D10 | D5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C20.14F5 ►in GL6(𝔽41)
| 22 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [22,9,0,0,0,0,32,0,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,18,0,0,0,0,0,0,16],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0] >;
C20.14F5 in GAP, Magma, Sage, TeX
C_{20}._{14}F_5 % in TeX
G:=Group("C20.14F5"); // GroupNames label
G:=SmallGroup(400,142);
// by ID
G=gap.SmallGroup(400,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,50,1924,8645,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=1,c^4=a^10,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=b^3>;
// generators/relations
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