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## G = C2×D4×D5order 160 = 25·5

### Direct product of C2, D4 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4×D5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — C2×D4×D5
 Lower central C5 — C10 — C2×D4×D5
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×D5
G = < a,b,c,d,e | a2=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 808 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, C10, C10, C10, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C2×D4×D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4×D5, C23×D5, C2×D4×D5

Smallest permutation representation of C2×D4×D5
On 40 points
Generators in S40
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 24 9 29)(2 25 10 30)(3 21 6 26)(4 22 7 27)(5 23 8 28)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 5A 5B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 1 1 2 2 2 2 5 5 5 5 10 10 10 10 2 2 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 C2 D4 D5 D10 D10 D10 D4×D5 kernel C2×D4×D5 C2×C4×D5 C2×D20 D4×D5 C2×C5⋊D4 D4×C10 C23×D5 D10 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 1 2 4 2 2 8 4 4

Matrix representation of C2×D4×D5 in GL4(𝔽41) generated by

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

C2×D4×D5 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_5
% in TeX

G:=Group("C2xD4xD5");
// GroupNames label

G:=SmallGroup(160,217);
// by ID

G=gap.SmallGroup(160,217);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,159,4613]);
// Polycyclic

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

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