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## G = C5×C22⋊C4order 80 = 24·5

### Direct product of C5 and C22⋊C4

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

Aliases: C5×C22⋊C4, C22⋊C20, C23.C10, C10.12D4, (C2×C20)⋊2C2, (C2×C10)⋊3C4, (C2×C4)⋊1C10, C2.1(C5×D4), C2.1(C2×C20), C10.17(C2×C4), (C22×C10).1C2, C22.2(C2×C10), (C2×C10).13C22, SmallGroup(80,21)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C22⋊C4
G = < a,b,c,d | a5=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C5×C22⋊C4
On 40 points
Generators in S40
(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)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 31)(25 32)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(21 28)(22 29)(23 30)(24 26)(25 27)
(1 6 30 36)(2 7 26 37)(3 8 27 38)(4 9 28 39)(5 10 29 40)(11 35 16 23)(12 31 17 24)(13 32 18 25)(14 33 19 21)(15 34 20 22)

G:=sub<Sym(40)| (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), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,31)(25,32), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,36)(12,37)(13,38)(14,39)(15,40)(21,28)(22,29)(23,30)(24,26)(25,27), (1,6,30,36)(2,7,26,37)(3,8,27,38)(4,9,28,39)(5,10,29,40)(11,35,16,23)(12,31,17,24)(13,32,18,25)(14,33,19,21)(15,34,20,22)>;

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), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,31)(25,32), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,36)(12,37)(13,38)(14,39)(15,40)(21,28)(22,29)(23,30)(24,26)(25,27), (1,6,30,36)(2,7,26,37)(3,8,27,38)(4,9,28,39)(5,10,29,40)(11,35,16,23)(12,31,17,24)(13,32,18,25)(14,33,19,21)(15,34,20,22) );

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)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,11),(7,12),(8,13),(9,14),(10,15),(16,36),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,31),(25,32)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,16),(7,17),(8,18),(9,19),(10,20),(11,36),(12,37),(13,38),(14,39),(15,40),(21,28),(22,29),(23,30),(24,26),(25,27)], [(1,6,30,36),(2,7,26,37),(3,8,27,38),(4,9,28,39),(5,10,29,40),(11,35,16,23),(12,31,17,24),(13,32,18,25),(14,33,19,21),(15,34,20,22)]])

C5×C22⋊C4 is a maximal subgroup of
C23.1D10  C23.11D10  Dic5.14D4  C23.D10  Dic54D4  C22⋊D20  D10.12D4  D10⋊D4  Dic5.5D4  C22.D20  D4×C20

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 20A ··· 20P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C4 C5 C10 C10 C20 D4 C5×D4 kernel C5×C22⋊C4 C2×C20 C22×C10 C2×C10 C22⋊C4 C2×C4 C23 C22 C10 C2 # reps 1 2 1 4 4 8 4 16 2 8

Matrix representation of C5×C22⋊C4 in GL3(𝔽41) generated by

 1 0 0 0 10 0 0 0 10
,
 1 0 0 0 1 0 0 0 40
,
 1 0 0 0 40 0 0 0 40
,
 9 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[1,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,40],[9,0,0,0,0,1,0,1,0] >;

C5×C22⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes C_4
% in TeX

G:=Group("C5xC2^2:C4");
// GroupNames label

G:=SmallGroup(80,21);
// by ID

G=gap.SmallGroup(80,21);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,200,221]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^2=d^4=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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