metabelian, supersoluble, monomial
Aliases: C20⋊5F5, D5.1D20, C20⋊1Dic5, Dic5⋊3Dic5, D10.12D10, D5.2Dic10, (C5×C20)⋊1C4, C4⋊(D5.D5), C5⋊(C4⋊Dic5), C5⋊4(C4⋊F5), C52⋊6(C4⋊C4), (C5×D5).4D4, (C4×D5).4D5, (C5×D5).3Q8, (C5×Dic5)⋊7C4, (D5×C20).3C2, C10.38(C2×F5), C10.4(C2×Dic5), (D5×C10).21C22, C2.5(C2×D5.D5), (C5×C10).23(C2×C4), (C2×D5.D5).3C2, SmallGroup(400,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊5F5
G = < a,b,c | a20=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
5C2
5C2
4C5
5C4
5C22
50C4
50C4
4C10
5C10
5C10
25C2×C4
25C2×C4
4C20
5C20
10F5
10Dic5
10F5
10Dic5
25C4⋊C4
Smallest permutation representation of C20⋊5F5
►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 73 47 21)(2 72 48 40)(3 71 49 39)(4 70 50 38)(5 69 51 37)(6 68 52 36)(7 67 53 35)(8 66 54 34)(9 65 55 33)(10 64 56 32)(11 63 57 31)(12 62 58 30)(13 61 59 29)(14 80 60 28)(15 79 41 27)(16 78 42 26)(17 77 43 25)(18 76 44 24)(19 75 45 23)(20 74 46 22)
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), (1,73,47,21)(2,72,48,40)(3,71,49,39)(4,70,50,38)(5,69,51,37)(6,68,52,36)(7,67,53,35)(8,66,54,34)(9,65,55,33)(10,64,56,32)(11,63,57,31)(12,62,58,30)(13,61,59,29)(14,80,60,28)(15,79,41,27)(16,78,42,26)(17,77,43,25)(18,76,44,24)(19,75,45,23)(20,74,46,22)>;
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), (1,73,47,21)(2,72,48,40)(3,71,49,39)(4,70,50,38)(5,69,51,37)(6,68,52,36)(7,67,53,35)(8,66,54,34)(9,65,55,33)(10,64,56,32)(11,63,57,31)(12,62,58,30)(13,61,59,29)(14,80,60,28)(15,79,41,27)(16,78,42,26)(17,77,43,25)(18,76,44,24)(19,75,45,23)(20,74,46,22) );
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)], [(1,73,47,21),(2,72,48,40),(3,71,49,39),(4,70,50,38),(5,69,51,37),(6,68,52,36),(7,67,53,35),(8,66,54,34),(9,65,55,33),(10,64,56,32),(11,63,57,31),(12,62,58,30),(13,61,59,29),(14,80,60,28),(15,79,41,27),(16,78,42,26),(17,77,43,25),(18,76,44,24),(19,75,45,23),(20,74,46,22)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | ··· | 5G | 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 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 5 | 5 | 2 | 10 | 50 | 50 | 50 | 50 | 2 | 2 | 4 | ··· | 4 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | - | + | - | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | Dic5 | Dic5 | D10 | Dic10 | D20 | F5 | C2×F5 | C4⋊F5 | D5.D5 | C2×D5.D5 | C20⋊5F5 |
| kernel | C20⋊5F5 | D5×C20 | C2×D5.D5 | C5×Dic5 | C5×C20 | C5×D5 | C5×D5 | C4×D5 | Dic5 | C20 | D10 | D5 | D5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C20⋊5F5 ►in GL6(𝔽41)
| 25 | 2 | 0 | 0 | 0 | 0 |
| 39 | 13 | 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 |
| 31 | 37 | 0 | 0 | 0 | 0 |
| 15 | 10 | 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))| [25,39,0,0,0,0,2,13,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[31,15,0,0,0,0,37,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0] >;
C20⋊5F5 in GAP, Magma, Sage, TeX
C_{20}\rtimes_5F_5 % in TeX
G:=Group("C20:5F5"); // GroupNames label
G:=SmallGroup(400,145);
// by ID
G=gap.SmallGroup(400,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,1924,8645,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
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