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## G = C5×C4○D4order 80 = 24·5

### Direct product of C5 and C4○D4

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

Aliases: C5×C4○D4, D4C20, Q8C20, D42C10, Q82C10, C20.21C22, C10.13C23, C4(C5×D4), C4(C5×Q8), C20(C5×D4), C20(C5×Q8), (C2×C20)⋊7C2, (C2×C4)⋊3C10, (C5×D4)⋊5C2, (C5×Q8)⋊5C2, C4.5(C2×C10), C22.(C2×C10), C2.3(C22×C10), (C2×C10).2C22, SmallGroup(80,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C4○D4
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — C5×C4○D4
 Lower central C1 — C2 — C5×C4○D4
 Upper central C1 — C20 — C5×C4○D4

Generators and relations for C5×C4○D4
G = < a,b,c,d | a5=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C5×C4○D4
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 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 31 40 30)(7 32 36 26)(8 33 37 27)(9 34 38 28)(10 35 39 29)
(1 17 13 23)(2 18 14 24)(3 19 15 25)(4 20 11 21)(5 16 12 22)(6 31 40 30)(7 32 36 26)(8 33 37 27)(9 34 38 28)(10 35 39 29)
(1 37)(2 38)(3 39)(4 40)(5 36)(6 11)(7 12)(8 13)(9 14)(10 15)(16 32)(17 33)(18 34)(19 35)(20 31)(21 30)(22 26)(23 27)(24 28)(25 29)

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

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

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

C5×C4○D4 is a maximal subgroup of   D42Dic5  D4.Dic5  D4⋊D10  D4.8D10  D4.9D10  D48D10  D4.10D10
C5×C4○D4 is a maximal quotient of   D4×C20  Q8×C20

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C5 C10 C10 C10 C4○D4 C5×C4○D4 kernel C5×C4○D4 C2×C20 C5×D4 C5×Q8 C4○D4 C2×C4 D4 Q8 C5 C1 # reps 1 3 3 1 4 12 12 4 2 8

Matrix representation of C5×C4○D4 in GL2(𝔽41) generated by

 16 0 0 16
,
 9 0 0 9
,
 32 13 0 9
,
 13 18 18 28
G:=sub<GL(2,GF(41))| [16,0,0,16],[9,0,0,9],[32,0,13,9],[13,18,18,28] >;

C5×C4○D4 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_4
% in TeX

G:=Group("C5xC4oD4");
// GroupNames label

G:=SmallGroup(80,48);
// by ID

G=gap.SmallGroup(80,48);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-2,421,162]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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