metabelian, supersoluble, monomial
Aliases: C20⋊1F5, (C5×C20)⋊3C4, C4⋊(C5⋊F5), C5⋊2(C4⋊F5), C5⋊D5.5D4, C5⋊D5.3Q8, C52⋊7(C4⋊C4), C52⋊6C4⋊7C4, C10.17(C2×F5), (C4×C5⋊D5).7C2, C2.5(C2×C5⋊F5), (C5×C10).30(C2×C4), (C2×C5⋊F5).3C2, (C2×C5⋊D5).22C22, SmallGroup(400,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊F5
G = < a,b,c | a20=b5=c4=1, ab=ba, cac-1=a3, cbc-1=b3 >
Subgroups: 696 in 104 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D5, C10, C4⋊C4, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4⋊F5, C52⋊6C4, C5×C20, C5⋊F5, C2×C5⋊D5, C4×C5⋊D5, C2×C5⋊F5, C20⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C4⋊C4, F5, C2×F5, C4⋊F5, C5⋊F5, C2×C5⋊F5, C20⋊F5
►On 100 points
Generators in S100
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 30 70 84 44)(2 31 71 85 45)(3 32 72 86 46)(4 33 73 87 47)(5 34 74 88 48)(6 35 75 89 49)(7 36 76 90 50)(8 37 77 91 51)(9 38 78 92 52)(10 39 79 93 53)(11 40 80 94 54)(12 21 61 95 55)(13 22 62 96 56)(14 23 63 97 57)(15 24 64 98 58)(16 25 65 99 59)(17 26 66 100 60)(18 27 67 81 41)(19 28 68 82 42)(20 29 69 83 43)
(2 8 10 4)(3 15 19 7)(5 9 17 13)(6 16)(12 18 20 14)(21 67 43 97)(22 74 52 100)(23 61 41 83)(24 68 50 86)(25 75 59 89)(26 62 48 92)(27 69 57 95)(28 76 46 98)(29 63 55 81)(30 70 44 84)(31 77 53 87)(32 64 42 90)(33 71 51 93)(34 78 60 96)(35 65 49 99)(36 72 58 82)(37 79 47 85)(38 66 56 88)(39 73 45 91)(40 80 54 94)
G:=sub<Sym(100)| (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,30,70,84,44)(2,31,71,85,45)(3,32,72,86,46)(4,33,73,87,47)(5,34,74,88,48)(6,35,75,89,49)(7,36,76,90,50)(8,37,77,91,51)(9,38,78,92,52)(10,39,79,93,53)(11,40,80,94,54)(12,21,61,95,55)(13,22,62,96,56)(14,23,63,97,57)(15,24,64,98,58)(16,25,65,99,59)(17,26,66,100,60)(18,27,67,81,41)(19,28,68,82,42)(20,29,69,83,43), (2,8,10,4)(3,15,19,7)(5,9,17,13)(6,16)(12,18,20,14)(21,67,43,97)(22,74,52,100)(23,61,41,83)(24,68,50,86)(25,75,59,89)(26,62,48,92)(27,69,57,95)(28,76,46,98)(29,63,55,81)(30,70,44,84)(31,77,53,87)(32,64,42,90)(33,71,51,93)(34,78,60,96)(35,65,49,99)(36,72,58,82)(37,79,47,85)(38,66,56,88)(39,73,45,91)(40,80,54,94)>;
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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,30,70,84,44)(2,31,71,85,45)(3,32,72,86,46)(4,33,73,87,47)(5,34,74,88,48)(6,35,75,89,49)(7,36,76,90,50)(8,37,77,91,51)(9,38,78,92,52)(10,39,79,93,53)(11,40,80,94,54)(12,21,61,95,55)(13,22,62,96,56)(14,23,63,97,57)(15,24,64,98,58)(16,25,65,99,59)(17,26,66,100,60)(18,27,67,81,41)(19,28,68,82,42)(20,29,69,83,43), (2,8,10,4)(3,15,19,7)(5,9,17,13)(6,16)(12,18,20,14)(21,67,43,97)(22,74,52,100)(23,61,41,83)(24,68,50,86)(25,75,59,89)(26,62,48,92)(27,69,57,95)(28,76,46,98)(29,63,55,81)(30,70,44,84)(31,77,53,87)(32,64,42,90)(33,71,51,93)(34,78,60,96)(35,65,49,99)(36,72,58,82)(37,79,47,85)(38,66,56,88)(39,73,45,91)(40,80,54,94) );
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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,30,70,84,44),(2,31,71,85,45),(3,32,72,86,46),(4,33,73,87,47),(5,34,74,88,48),(6,35,75,89,49),(7,36,76,90,50),(8,37,77,91,51),(9,38,78,92,52),(10,39,79,93,53),(11,40,80,94,54),(12,21,61,95,55),(13,22,62,96,56),(14,23,63,97,57),(15,24,64,98,58),(16,25,65,99,59),(17,26,66,100,60),(18,27,67,81,41),(19,28,68,82,42),(20,29,69,83,43)], [(2,8,10,4),(3,15,19,7),(5,9,17,13),(6,16),(12,18,20,14),(21,67,43,97),(22,74,52,100),(23,61,41,83),(24,68,50,86),(25,75,59,89),(26,62,48,92),(27,69,57,95),(28,76,46,98),(29,63,55,81),(30,70,44,84),(31,77,53,87),(32,64,42,90),(33,71,51,93),(34,78,60,96),(35,65,49,99),(36,72,58,82),(37,79,47,85),(38,66,56,88),(39,73,45,91),(40,80,54,94)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | ··· | 4F | 5A | ··· | 5F | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 25 | 25 | 2 | 50 | ··· | 50 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | F5 | C2×F5 | C4⋊F5 |
| kernel | C20⋊F5 | C4×C5⋊D5 | C2×C5⋊F5 | C52⋊6C4 | C5×C20 | C5⋊D5 | C5⋊D5 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 6 | 6 | 12 |
Matrix representation of C20⋊F5 ►in GL8(𝔽41)
| 40 | 40 | 40 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 | 7 | 34 |
| 0 | 0 | 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 34 | 27 |
| 0 | 0 | 0 | 0 | 14 | 34 | 7 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(41))| [40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,14,7,14,0,0,0,0,27,27,0,34,0,0,0,0,7,34,34,7,0,0,0,0,34,0,27,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
C20⋊F5 in GAP, Magma, Sage, TeX
C_{20}\rtimes F_5 % in TeX
G:=Group("C20:F5"); // GroupNames label
G:=SmallGroup(400,152);
// by ID
G=gap.SmallGroup(400,152);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,964,496,5765,2897]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^3>;
// generators/relations