direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊C4, C4⋊C20, C20⋊5C4, C10.3Q8, C10.13D4, C2.(C5×Q8), C2.2(C5×D4), (C2×C20).2C2, C2.2(C2×C20), (C2×C4).1C10, C10.18(C2×C4), C22.3(C2×C10), (C2×C10).14C22, SmallGroup(80,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊C4
G = < a,b,c | a5=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C4⋊C4
►Regular action 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 51 26 41)(2 52 27 42)(3 53 28 43)(4 54 29 44)(5 55 30 45)(6 66 16 56)(7 67 17 57)(8 68 18 58)(9 69 19 59)(10 70 20 60)(11 61 76 71)(12 62 77 72)(13 63 78 73)(14 64 79 74)(15 65 80 75)(21 46 31 36)(22 47 32 37)(23 48 33 38)(24 49 34 39)(25 50 35 40)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 41 76 36)(7 42 77 37)(8 43 78 38)(9 44 79 39)(10 45 80 40)(11 46 16 51)(12 47 17 52)(13 48 18 53)(14 49 19 54)(15 50 20 55)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
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), (1,51,26,41)(2,52,27,42)(3,53,28,43)(4,54,29,44)(5,55,30,45)(6,66,16,56)(7,67,17,57)(8,68,18,58)(9,69,19,59)(10,70,20,60)(11,61,76,71)(12,62,77,72)(13,63,78,73)(14,64,79,74)(15,65,80,75)(21,46,31,36)(22,47,32,37)(23,48,33,38)(24,49,34,39)(25,50,35,40), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,41,76,36)(7,42,77,37)(8,43,78,38)(9,44,79,39)(10,45,80,40)(11,46,16,51)(12,47,17,52)(13,48,18,53)(14,49,19,54)(15,50,20,55)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)>;
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), (1,51,26,41)(2,52,27,42)(3,53,28,43)(4,54,29,44)(5,55,30,45)(6,66,16,56)(7,67,17,57)(8,68,18,58)(9,69,19,59)(10,70,20,60)(11,61,76,71)(12,62,77,72)(13,63,78,73)(14,64,79,74)(15,65,80,75)(21,46,31,36)(22,47,32,37)(23,48,33,38)(24,49,34,39)(25,50,35,40), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,41,76,36)(7,42,77,37)(8,43,78,38)(9,44,79,39)(10,45,80,40)(11,46,16,51)(12,47,17,52)(13,48,18,53)(14,49,19,54)(15,50,20,55)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70) );
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)], [(1,51,26,41),(2,52,27,42),(3,53,28,43),(4,54,29,44),(5,55,30,45),(6,66,16,56),(7,67,17,57),(8,68,18,58),(9,69,19,59),(10,70,20,60),(11,61,76,71),(12,62,77,72),(13,63,78,73),(14,64,79,74),(15,65,80,75),(21,46,31,36),(22,47,32,37),(23,48,33,38),(24,49,34,39),(25,50,35,40)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,41,76,36),(7,42,77,37),(8,43,78,38),(9,44,79,39),(10,45,80,40),(11,46,16,51),(12,47,17,52),(13,48,18,53),(14,49,19,54),(15,50,20,55),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70)]])
C5×C4⋊C4 is a maximal subgroup of
C10.D8 C20.Q8 D20⋊6C4 C10.Q16 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊Q8 D10⋊2Q8 C4⋊C4⋊D5 D4×C20 Q8×C20
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C5×C4⋊C4 | C2×C20 | C20 | C4⋊C4 | C2×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 3 | 4 | 4 | 12 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C4⋊C4 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 9 | 0 | 0 |
| 0 | 27 | 16 |
| 0 | 16 | 14 |
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[1,0,0,0,0,40,0,1,0],[9,0,0,0,27,16,0,16,14] >;
C5×C4⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes C_4
% in TeX
G:=Group("C5xC4:C4"); // GroupNames label
G:=SmallGroup(80,22);
// by ID
G=gap.SmallGroup(80,22);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-2,200,221,106]);
// Polycyclic
G:=Group<a,b,c|a^5=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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