direct product, metabelian, nilpotent (class 2), monomial, 5-elementary
Aliases: C4×He5, C52⋊3C20, C20.1C52, (C5×C20)⋊C5, C2.(C2×He5), C5.1(C5×C20), C10.2(C5×C10), (C5×C10).2C10, (C2×He5).3C2, SmallGroup(500,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×He5
G = < a,b,c,d | a4=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C4×He5
►On 100 points
Generators in S100
(1 21 62 13)(2 22 63 14)(3 23 64 15)(4 24 65 11)(5 25 61 12)(6 77 46 59)(7 78 47 60)(8 79 48 56)(9 80 49 57)(10 76 50 58)(16 68 27 91)(17 69 28 92)(18 70 29 93)(19 66 30 94)(20 67 26 95)(31 73 42 85)(32 74 43 81)(33 75 44 82)(34 71 45 83)(35 72 41 84)(36 97 54 86)(37 98 55 87)(38 99 51 88)(39 100 52 89)(40 96 53 90)
(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 71 49 36 29)(2 72 50 37 30)(3 73 46 38 26)(4 74 47 39 27)(5 75 48 40 28)(6 51 20 64 85)(7 52 16 65 81)(8 53 17 61 82)(9 54 18 62 83)(10 55 19 63 84)(11 32 78 89 68)(12 33 79 90 69)(13 34 80 86 70)(14 35 76 87 66)(15 31 77 88 67)(21 45 57 97 93)(22 41 58 98 94)(23 42 59 99 95)(24 43 60 100 91)(25 44 56 96 92)
(1 2 26 47 28)(3 39 5 71 72)(4 48 36 37 46)(6 65 8 54 55)(7 17 62 63 20)(9 10 85 16 82)(11 79 86 87 77)(12 34 35 15 89)(13 14 67 78 69)(18 19 51 81 53)(21 22 95 60 92)(23 100 25 45 41)(24 56 97 98 59)(27 75 49 50 73)(29 30 38 74 40)(31 68 33 80 76)(32 90 70 66 88)(42 91 44 57 58)(43 96 93 94 99)(52 61 83 84 64)
G:=sub<Sym(100)| (1,21,62,13)(2,22,63,14)(3,23,64,15)(4,24,65,11)(5,25,61,12)(6,77,46,59)(7,78,47,60)(8,79,48,56)(9,80,49,57)(10,76,50,58)(16,68,27,91)(17,69,28,92)(18,70,29,93)(19,66,30,94)(20,67,26,95)(31,73,42,85)(32,74,43,81)(33,75,44,82)(34,71,45,83)(35,72,41,84)(36,97,54,86)(37,98,55,87)(38,99,51,88)(39,100,52,89)(40,96,53,90), (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), (1,71,49,36,29)(2,72,50,37,30)(3,73,46,38,26)(4,74,47,39,27)(5,75,48,40,28)(6,51,20,64,85)(7,52,16,65,81)(8,53,17,61,82)(9,54,18,62,83)(10,55,19,63,84)(11,32,78,89,68)(12,33,79,90,69)(13,34,80,86,70)(14,35,76,87,66)(15,31,77,88,67)(21,45,57,97,93)(22,41,58,98,94)(23,42,59,99,95)(24,43,60,100,91)(25,44,56,96,92), (1,2,26,47,28)(3,39,5,71,72)(4,48,36,37,46)(6,65,8,54,55)(7,17,62,63,20)(9,10,85,16,82)(11,79,86,87,77)(12,34,35,15,89)(13,14,67,78,69)(18,19,51,81,53)(21,22,95,60,92)(23,100,25,45,41)(24,56,97,98,59)(27,75,49,50,73)(29,30,38,74,40)(31,68,33,80,76)(32,90,70,66,88)(42,91,44,57,58)(43,96,93,94,99)(52,61,83,84,64)>;
G:=Group( (1,21,62,13)(2,22,63,14)(3,23,64,15)(4,24,65,11)(5,25,61,12)(6,77,46,59)(7,78,47,60)(8,79,48,56)(9,80,49,57)(10,76,50,58)(16,68,27,91)(17,69,28,92)(18,70,29,93)(19,66,30,94)(20,67,26,95)(31,73,42,85)(32,74,43,81)(33,75,44,82)(34,71,45,83)(35,72,41,84)(36,97,54,86)(37,98,55,87)(38,99,51,88)(39,100,52,89)(40,96,53,90), (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), (1,71,49,36,29)(2,72,50,37,30)(3,73,46,38,26)(4,74,47,39,27)(5,75,48,40,28)(6,51,20,64,85)(7,52,16,65,81)(8,53,17,61,82)(9,54,18,62,83)(10,55,19,63,84)(11,32,78,89,68)(12,33,79,90,69)(13,34,80,86,70)(14,35,76,87,66)(15,31,77,88,67)(21,45,57,97,93)(22,41,58,98,94)(23,42,59,99,95)(24,43,60,100,91)(25,44,56,96,92), (1,2,26,47,28)(3,39,5,71,72)(4,48,36,37,46)(6,65,8,54,55)(7,17,62,63,20)(9,10,85,16,82)(11,79,86,87,77)(12,34,35,15,89)(13,14,67,78,69)(18,19,51,81,53)(21,22,95,60,92)(23,100,25,45,41)(24,56,97,98,59)(27,75,49,50,73)(29,30,38,74,40)(31,68,33,80,76)(32,90,70,66,88)(42,91,44,57,58)(43,96,93,94,99)(52,61,83,84,64) );
G=PermutationGroup([[(1,21,62,13),(2,22,63,14),(3,23,64,15),(4,24,65,11),(5,25,61,12),(6,77,46,59),(7,78,47,60),(8,79,48,56),(9,80,49,57),(10,76,50,58),(16,68,27,91),(17,69,28,92),(18,70,29,93),(19,66,30,94),(20,67,26,95),(31,73,42,85),(32,74,43,81),(33,75,44,82),(34,71,45,83),(35,72,41,84),(36,97,54,86),(37,98,55,87),(38,99,51,88),(39,100,52,89),(40,96,53,90)], [(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)], [(1,71,49,36,29),(2,72,50,37,30),(3,73,46,38,26),(4,74,47,39,27),(5,75,48,40,28),(6,51,20,64,85),(7,52,16,65,81),(8,53,17,61,82),(9,54,18,62,83),(10,55,19,63,84),(11,32,78,89,68),(12,33,79,90,69),(13,34,80,86,70),(14,35,76,87,66),(15,31,77,88,67),(21,45,57,97,93),(22,41,58,98,94),(23,42,59,99,95),(24,43,60,100,91),(25,44,56,96,92)], [(1,2,26,47,28),(3,39,5,71,72),(4,48,36,37,46),(6,65,8,54,55),(7,17,62,63,20),(9,10,85,16,82),(11,79,86,87,77),(12,34,35,15,89),(13,14,67,78,69),(18,19,51,81,53),(21,22,95,60,92),(23,100,25,45,41),(24,56,97,98,59),(27,75,49,50,73),(29,30,38,74,40),(31,68,33,80,76),(32,90,70,66,88),(42,91,44,57,58),(43,96,93,94,99),(52,61,83,84,64)]])
116 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5AB | 10A | 10B | 10C | 10D | 10E | ··· | 10AB | 20A | ··· | 20H | 20I | ··· | 20BD |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 1 | ··· | 1 | 5 | ··· | 5 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 |
| type | + | + | |||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | He5 | C2×He5 | C4×He5 |
| kernel | C4×He5 | C2×He5 | He5 | C5×C20 | C5×C10 | C52 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 24 | 24 | 48 | 4 | 4 | 8 |
Matrix representation of C4×He5 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 38 | 10 | 30 | 9 | 7 |
| 0 | 4 | 0 | 11 | 32 | 34 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 0 | 9 | 0 | 10 | 28 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 38 | 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 10 | 28 | 3 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[16,0,0,0,0,0,0,38,4,0,0,9,0,10,0,0,0,0,0,30,11,0,0,10,0,9,32,16,0,28,0,7,34,0,37,3],[1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[18,0,0,0,0,0,0,38,4,0,0,9,0,1,0,0,0,0,0,0,0,0,0,10,0,0,0,1,0,28,0,0,0,0,1,3] >;
C4×He5 in GAP, Magma, Sage, TeX
C_4\times {\rm He}_5 % in TeX
G:=Group("C4xHe5"); // GroupNames label
G:=SmallGroup(500,13);
// by ID
G=gap.SmallGroup(500,13);
# by ID
G:=PCGroup([5,-2,-5,-5,-2,-5,250,832]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^5=c^5=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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