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## G = C5×C5⋊Q16order 400 = 24·52

### Direct product of C5 and C5⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C5×C5⋊Q16
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×Dic10 — C5×C5⋊Q16
 Lower central C5 — C10 — C20 — C5×C5⋊Q16
 Upper central C1 — C10 — C20 — C5×Q8

Generators and relations for C5×C5⋊Q16
G = < a,b,c,d | a5=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Smallest permutation representation of C5×C5⋊Q16
On 80 points
Generators in S80
(1 60 38 12 66)(2 61 39 13 67)(3 62 40 14 68)(4 63 33 15 69)(5 64 34 16 70)(6 57 35 9 71)(7 58 36 10 72)(8 59 37 11 65)(17 42 50 75 29)(18 43 51 76 30)(19 44 52 77 31)(20 45 53 78 32)(21 46 54 79 25)(22 47 55 80 26)(23 48 56 73 27)(24 41 49 74 28)
(1 60 38 12 66)(2 67 13 39 61)(3 62 40 14 68)(4 69 15 33 63)(5 64 34 16 70)(6 71 9 35 57)(7 58 36 10 72)(8 65 11 37 59)(17 42 50 75 29)(18 30 76 51 43)(19 44 52 77 31)(20 32 78 53 45)(21 46 54 79 25)(22 26 80 55 47)(23 48 56 73 27)(24 28 74 49 41)
(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)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 75 5 79)(2 74 6 78)(3 73 7 77)(4 80 8 76)(9 45 13 41)(10 44 14 48)(11 43 15 47)(12 42 16 46)(17 34 21 38)(18 33 22 37)(19 40 23 36)(20 39 24 35)(25 60 29 64)(26 59 30 63)(27 58 31 62)(28 57 32 61)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)

G:=sub<Sym(80)| (1,60,38,12,66)(2,61,39,13,67)(3,62,40,14,68)(4,63,33,15,69)(5,64,34,16,70)(6,57,35,9,71)(7,58,36,10,72)(8,59,37,11,65)(17,42,50,75,29)(18,43,51,76,30)(19,44,52,77,31)(20,45,53,78,32)(21,46,54,79,25)(22,47,55,80,26)(23,48,56,73,27)(24,41,49,74,28), (1,60,38,12,66)(2,67,13,39,61)(3,62,40,14,68)(4,69,15,33,63)(5,64,34,16,70)(6,71,9,35,57)(7,58,36,10,72)(8,65,11,37,59)(17,42,50,75,29)(18,30,76,51,43)(19,44,52,77,31)(20,32,78,53,45)(21,46,54,79,25)(22,26,80,55,47)(23,48,56,73,27)(24,28,74,49,41), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,45,13,41)(10,44,14,48)(11,43,15,47)(12,42,16,46)(17,34,21,38)(18,33,22,37)(19,40,23,36)(20,39,24,35)(25,60,29,64)(26,59,30,63)(27,58,31,62)(28,57,32,61)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)>;

G:=Group( (1,60,38,12,66)(2,61,39,13,67)(3,62,40,14,68)(4,63,33,15,69)(5,64,34,16,70)(6,57,35,9,71)(7,58,36,10,72)(8,59,37,11,65)(17,42,50,75,29)(18,43,51,76,30)(19,44,52,77,31)(20,45,53,78,32)(21,46,54,79,25)(22,47,55,80,26)(23,48,56,73,27)(24,41,49,74,28), (1,60,38,12,66)(2,67,13,39,61)(3,62,40,14,68)(4,69,15,33,63)(5,64,34,16,70)(6,71,9,35,57)(7,58,36,10,72)(8,65,11,37,59)(17,42,50,75,29)(18,30,76,51,43)(19,44,52,77,31)(20,32,78,53,45)(21,46,54,79,25)(22,26,80,55,47)(23,48,56,73,27)(24,28,74,49,41), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,45,13,41)(10,44,14,48)(11,43,15,47)(12,42,16,46)(17,34,21,38)(18,33,22,37)(19,40,23,36)(20,39,24,35)(25,60,29,64)(26,59,30,63)(27,58,31,62)(28,57,32,61)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72) );

G=PermutationGroup([[(1,60,38,12,66),(2,61,39,13,67),(3,62,40,14,68),(4,63,33,15,69),(5,64,34,16,70),(6,57,35,9,71),(7,58,36,10,72),(8,59,37,11,65),(17,42,50,75,29),(18,43,51,76,30),(19,44,52,77,31),(20,45,53,78,32),(21,46,54,79,25),(22,47,55,80,26),(23,48,56,73,27),(24,41,49,74,28)], [(1,60,38,12,66),(2,67,13,39,61),(3,62,40,14,68),(4,69,15,33,63),(5,64,34,16,70),(6,71,9,35,57),(7,58,36,10,72),(8,65,11,37,59),(17,42,50,75,29),(18,30,76,51,43),(19,44,52,77,31),(20,32,78,53,45),(21,46,54,79,25),(22,26,80,55,47),(23,48,56,73,27),(24,28,74,49,41)], [(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),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,75,5,79),(2,74,6,78),(3,73,7,77),(4,80,8,76),(9,45,13,41),(10,44,14,48),(11,43,15,47),(12,42,16,46),(17,34,21,38),(18,33,22,37),(19,40,23,36),(20,39,24,35),(25,60,29,64),(26,59,30,63),(27,58,31,62),(28,57,32,61),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)]])

85 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 5C 5D 5E ··· 5N 8A 8B 10A 10B 10C 10D 10E ··· 10N 20A 20B 20C 20D 20E ··· 20AL 20AM 20AN 20AO 20AP 40A ··· 40H order 1 2 4 4 4 5 5 5 5 5 ··· 5 8 8 10 10 10 10 10 ··· 10 20 20 20 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 20 1 1 1 1 2 ··· 2 10 10 1 1 1 1 2 ··· 2 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 Q16 D10 C5⋊D4 C5×D4 C5×D5 C5×Q16 D5×C10 C5×C5⋊D4 C5⋊Q16 C5×C5⋊Q16 kernel C5×C5⋊Q16 C5×C5⋊2C8 C5×Dic10 Q8×C52 C5⋊Q16 C5⋊2C8 Dic10 C5×Q8 C5×C10 C5×Q8 C52 C20 C10 C10 Q8 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×C5⋊Q16 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 0 40 0 0 40 0 0 0 0 0 0 12 0 0 17 17
,
 40 0 0 0 0 40 0 0 0 0 34 7 0 0 28 7
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[10,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,0,17,0,0,12,17],[40,0,0,0,0,40,0,0,0,0,34,28,0,0,7,7] >;

C5×C5⋊Q16 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C5xC5:Q16");
// GroupNames label

G:=SmallGroup(400,90);
// by ID

G=gap.SmallGroup(400,90);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,240,265,247,1443,729,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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