metabelian, supersoluble, monomial
Aliases: C20.12F5, C20.1Dic5, C52:6M4(2), D10.3Dic5, Dic5.13D10, (C5xC20).1C4, (C4xD5).3D5, C4.(D5.D5), C5:5(C4.F5), C52:3C8:5C2, (D5xC20).2C2, C10.36(C2xF5), (D5xC10).11C4, C5:1(C4.Dic5), C10.2(C2xDic5), (C5xDic5).17C22, C2.4(C2xD5.D5), (C5xC10).21(C2xC4), SmallGroup(400,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.12F5
G = < a,b,c | a20=b5=1, c4=a10, ab=ba, cac-1=a-1, cbc-1=b3 >
Quotients: C1, C2, C4, C22, C2xC4, D5, M4(2), Dic5, F5, D10, C2xDic5, C2xF5, C4.Dic5, C4.F5, D5.D5, C2xD5.D5, C20.12F5
Smallest permutation representation of C20.12F5
►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 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)(41 53 45 57 49)(42 54 46 58 50)(43 55 47 59 51)(44 56 48 60 52)(61 69 77 65 73)(62 70 78 66 74)(63 71 79 67 75)(64 72 80 68 76)
(1 77 25 51 11 67 35 41)(2 76 26 50 12 66 36 60)(3 75 27 49 13 65 37 59)(4 74 28 48 14 64 38 58)(5 73 29 47 15 63 39 57)(6 72 30 46 16 62 40 56)(7 71 31 45 17 61 21 55)(8 70 32 44 18 80 22 54)(9 69 33 43 19 79 23 53)(10 68 34 42 20 78 24 52)
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)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,77,25,51,11,67,35,41)(2,76,26,50,12,66,36,60)(3,75,27,49,13,65,37,59)(4,74,28,48,14,64,38,58)(5,73,29,47,15,63,39,57)(6,72,30,46,16,62,40,56)(7,71,31,45,17,61,21,55)(8,70,32,44,18,80,22,54)(9,69,33,43,19,79,23,53)(10,68,34,42,20,78,24,52)>;
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)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,69,77,65,73)(62,70,78,66,74)(63,71,79,67,75)(64,72,80,68,76), (1,77,25,51,11,67,35,41)(2,76,26,50,12,66,36,60)(3,75,27,49,13,65,37,59)(4,74,28,48,14,64,38,58)(5,73,29,47,15,63,39,57)(6,72,30,46,16,62,40,56)(7,71,31,45,17,61,21,55)(8,70,32,44,18,80,22,54)(9,69,33,43,19,79,23,53)(10,68,34,42,20,78,24,52) );
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),(21,37,33,29,25),(22,38,34,30,26),(23,39,35,31,27),(24,40,36,32,28),(41,53,45,57,49),(42,54,46,58,50),(43,55,47,59,51),(44,56,48,60,52),(61,69,77,65,73),(62,70,78,66,74),(63,71,79,67,75),(64,72,80,68,76)], [(1,77,25,51,11,67,35,41),(2,76,26,50,12,66,36,60),(3,75,27,49,13,65,37,59),(4,74,28,48,14,64,38,58),(5,73,29,47,15,63,39,57),(6,72,30,46,16,62,40,56),(7,71,31,45,17,61,21,55),(8,70,32,44,18,80,22,54),(9,69,33,43,19,79,23,53),(10,68,34,42,20,78,24,52)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | ··· | 5G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10G | 10H | 10I | 10J | 10K | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 10 | 2 | 5 | 5 | 2 | 2 | 4 | ··· | 4 | 50 | 50 | 50 | 50 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | D5 | M4(2) | D10 | Dic5 | Dic5 | C4.Dic5 | F5 | C2xF5 | C4.F5 | D5.D5 | C2xD5.D5 | C20.12F5 |
| kernel | C20.12F5 | C52:3C8 | D5xC20 | C5xC20 | D5xC10 | C4xD5 | C52 | Dic5 | C20 | D10 | C5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C20.12F5 ►in GL4(F41) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 39 | 0 |
| 0 | 0 | 0 | 39 |
| 18 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
G:=sub<GL(4,GF(41))| [20,0,0,0,0,20,0,0,0,0,39,0,0,0,0,39],[18,0,0,0,0,16,0,0,0,0,10,0,0,0,0,37],[0,0,40,0,0,0,0,40,0,40,0,0,1,0,0,0] >;
C20.12F5 in GAP, Magma, Sage, TeX
C_{20}._{12}F_5 % in TeX
G:=Group("C20.12F5"); // GroupNames label
G:=SmallGroup(400,143);
// by ID
G=gap.SmallGroup(400,143);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,1924,8645,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=1,c^4=a^10,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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