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## G = C5×C4⋊C4order 80 = 24·5

### Direct product of C5 and C4⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C4⋊C4, C4⋊C20, C205C4, 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

 Derived series C1 — C2 — C5×C4⋊C4
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4
 Lower central C1 — C2 — C5×C4⋊C4
 Upper central C1 — C2×C10 — C5×C4⋊C4

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  D206C4  C10.Q16  Dic53Q8  C20⋊Q8  Dic5.Q8  C4.Dic10  C4⋊C47D5  D208C4  D10.13D4  C4⋊D20  D10⋊Q8  D102Q8  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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