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## G = C5×D4.D5order 400 = 24·52

### Direct product of C5 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C5×D4.D5
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×Dic10 — C5×D4.D5
 Lower central C5 — C10 — C20 — C5×D4.D5
 Upper central C1 — C10 — C20 — C5×D4

Generators and relations for C5×D4.D5
G = < a,b,c,d,e | a5=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 140 in 56 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, D4, Q8, C10, C10, SD16, Dic5, C20, C20, C2×C10, C52, C52C8, C40, Dic10, C5×D4, C5×D4, C5×Q8, C5×C10, C5×C10, D4.D5, C5×SD16, C5×Dic5, C5×C20, C102, C5×C52C8, C5×Dic10, D4×C52, C5×D4.D5
Quotients: C1, C2, C22, C5, D4, D5, C10, SD16, D10, C2×C10, C5⋊D4, C5×D4, C5×D5, D4.D5, C5×SD16, D5×C10, C5×C5⋊D4, C5×D4.D5

Smallest permutation representation of C5×D4.D5
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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 35 34 33 32)(36 40 39 38 37)
(1 27 6 22)(2 28 7 23)(3 29 8 24)(4 30 9 25)(5 26 10 21)(11 37 16 32)(12 38 17 33)(13 39 18 34)(14 40 19 35)(15 36 20 31)

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

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

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

85 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 5E ··· 5N 8A 8B 10A 10B 10C 10D 10E ··· 10N 10O ··· 10AL 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R 40A ··· 40H order 1 2 2 4 4 5 5 5 5 5 ··· 5 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 20 20 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 4 2 20 1 1 1 1 2 ··· 2 10 10 1 1 1 1 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 20 20 20 20 10 ··· 10

85 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C5 C10 C10 C10 D4 D5 SD16 D10 C5⋊D4 C5×D4 C5×D5 C5×SD16 D5×C10 C5×C5⋊D4 D4.D5 C5×D4.D5 kernel C5×D4.D5 C5×C5⋊2C8 C5×Dic10 D4×C52 D4.D5 C5⋊2C8 Dic10 C5×D4 C5×C10 C5×D4 C52 C20 C10 C10 D4 C5 C4 C2 C5 C1 # reps 1 1 1 1 4 4 4 4 1 2 2 2 4 4 8 8 8 16 2 8

Matrix representation of C5×D4.D5 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 25 0 0 36 1
,
 40 0 0 0 0 1 0 0 0 0 1 0 0 0 5 40
,
 37 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 10 0 0 37 0 0 0 0 0 0 35 0 0 7 0
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,36,0,0,25,1],[40,0,0,0,0,1,0,0,0,0,1,5,0,0,0,40],[37,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,37,0,0,10,0,0,0,0,0,0,7,0,0,35,0] >;

C5×D4.D5 in GAP, Magma, Sage, TeX

C_5\times D_4.D_5
% in TeX

G:=Group("C5xD4.D5");
// GroupNames label

G:=SmallGroup(400,88);
// by ID

G=gap.SmallGroup(400,88);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,1443,729,69,11525]);
// Polycyclic

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

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