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## G = D5×C22⋊C4order 160 = 25·5

### Direct product of D5 and C22⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×C22⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4
 Lower central C5 — C10 — D5×C22⋊C4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D5×C22⋊C4
G = < a,b,c,d,e | a5=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 472 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], D5 [×4], D5 [×2], C10, C10 [×2], C10 [×2], C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×8], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, D5×C22⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4

Smallest permutation representation of D5×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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 1 1 2 2 5 5 5 5 10 10 2 2 2 2 10 10 10 10 2 2 2 ··· 2 4 4 4 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D5 D10 D10 C4×D5 D4×D5 kernel D5×C22⋊C4 D10⋊C4 C23.D5 C5×C22⋊C4 C2×C4×D5 C23×D5 C22×D5 D10 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 1 1 2 1 8 4 2 4 2 8 4

Matrix representation of D5×C22⋊C4 in GL4(𝔽41) generated by

 0 1 0 0 40 6 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 32 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 9 0 0 0 0 9 0 0 0 0 32 39 0 0 40 9
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,32,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,32,40,0,0,39,9] >;

D5×C22⋊C4 in GAP, Magma, Sage, TeX

D_5\times C_2^2\rtimes C_4
% in TeX

G:=Group("D5xC2^2:C4");
// GroupNames label

G:=SmallGroup(160,101);
// by ID

G=gap.SmallGroup(160,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,50,4613]);
// Polycyclic

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

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