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## G = C5×Q8⋊D5order 400 = 24·52

### Direct product of C5 and Q8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C5×Q8⋊D5
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×D20 — C5×Q8⋊D5
 Lower central C5 — C10 — C20 — C5×Q8⋊D5
 Upper central C1 — C10 — C20 — C5×Q8

Generators and relations for C5×Q8⋊D5
G = < a,b,c,d,e | a5=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Smallest permutation representation of C5×Q8⋊D5
On 80 points
Generators in S80
(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 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)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 76 46 71)(42 77 47 72)(43 78 48 73)(44 79 49 74)(45 80 50 75)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(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 45 44 43 42)(46 50 49 48 47)(51 55 54 53 52)(56 60 59 58 57)(61 65 64 63 62)(66 70 69 68 67)(71 75 74 73 72)(76 80 79 78 77)
(1 42)(2 43)(3 44)(4 45)(5 41)(6 47)(7 48)(8 49)(9 50)(10 46)(11 52)(12 53)(13 54)(14 55)(15 51)(16 57)(17 58)(18 59)(19 60)(20 56)(21 62)(22 63)(23 64)(24 65)(25 61)(26 67)(27 68)(28 69)(29 70)(30 66)(31 72)(32 73)(33 74)(34 75)(35 71)(36 77)(37 78)(38 79)(39 80)(40 76)

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

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

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

85 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 5E ··· 5N 8A 8B 10A 10B 10C 10D 10E ··· 10N 10O 10P 10Q 10R 20A 20B 20C 20D 20E ··· 20AL 40A ··· 40H order 1 2 2 4 4 5 5 5 5 5 ··· 5 8 8 10 10 10 10 10 ··· 10 10 10 10 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 20 2 4 1 1 1 1 2 ··· 2 10 10 1 1 1 1 2 ··· 2 20 20 20 20 2 2 2 2 4 ··· 4 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 Q8⋊D5 C5×Q8⋊D5 kernel C5×Q8⋊D5 C5×C5⋊2C8 C5×D20 Q8×C52 Q8⋊D5 C5⋊2C8 D20 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×Q8⋊D5 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 23 0 0 32 40
,
 1 0 0 0 0 1 0 0 0 0 23 28 0 0 25 18
,
 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 8 39 0 0 11 33
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,32,0,0,23,40],[1,0,0,0,0,1,0,0,0,0,23,25,0,0,28,18],[37,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,37,0,0,10,0,0,0,0,0,8,11,0,0,39,33] >;

C5×Q8⋊D5 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes D_5
% in TeX

G:=Group("C5xQ8:D5");
// GroupNames label

G:=SmallGroup(400,89);
// by ID

G=gap.SmallGroup(400,89);
# by ID

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

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

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