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## G = C5×Q8⋊2D5order 400 = 24·52

### Direct product of C5 and Q8⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×Q8⋊2D5
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C20 — C5×Q8⋊2D5
 Lower central C5 — C10 — C5×Q8⋊2D5
 Upper central C1 — C10 — C5×Q8

Generators and relations for C5×Q82D5
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, ce=ec, ede=d-1 >

Subgroups: 252 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, C20, D10, C2×C10, C52, C4×D5, D20, C2×C20, C5×D4, C5×Q8, C5×Q8, C5×D5, C5×C10, Q82D5, C5×C4○D4, C5×Dic5, C5×C20, D5×C10, D5×C20, C5×D20, Q8×C52, C5×Q82D5
Quotients: C1, C2, C22, C5, C23, D5, C10, C4○D4, D10, C2×C10, C22×D5, C22×C10, C5×D5, Q82D5, C5×C4○D4, D5×C10, D5×C2×C10, C5×Q82D5

Smallest permutation representation of C5×Q82D5
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 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(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 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 4 2 5 3)(6 9 7 10 8)(11 14 12 15 13)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)(31 34 32 35 33)(36 39 37 40 38)(41 43 45 42 44)(46 48 50 47 49)(51 53 55 52 54)(56 58 60 57 59)(61 63 65 62 64)(66 68 70 67 69)(71 73 75 72 74)(76 78 80 77 79)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 49)(7 50)(8 46)(9 47)(10 48)(11 54)(12 55)(13 51)(14 52)(15 53)(16 59)(17 60)(18 56)(19 57)(20 58)(21 64)(22 65)(23 61)(24 62)(25 63)(26 69)(27 70)(28 66)(29 67)(30 68)(31 74)(32 75)(33 71)(34 72)(35 73)(36 79)(37 80)(38 76)(39 77)(40 78)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10Z 20A ··· 20L 20M ··· 20AP 20AQ ··· 20AX order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 10 10 10 2 2 2 5 5 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 10 ··· 10 2 ··· 2 4 ··· 4 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D5 C4○D4 D10 C5×D5 C5×C4○D4 D5×C10 Q8⋊2D5 C5×Q8⋊2D5 kernel C5×Q8⋊2D5 D5×C20 C5×D20 Q8×C52 Q8⋊2D5 C4×D5 D20 C5×Q8 C5×Q8 C52 C20 Q8 C5 C4 C5 C1 # reps 1 3 3 1 4 12 12 4 2 2 6 8 8 24 2 8

Matrix representation of C5×Q82D5 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 5 0 0 16 40
,
 1 0 0 0 0 1 0 0 0 0 4 30 0 0 9 37
,
 37 7 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 10 34 0 0 20 31 0 0 0 0 36 24 0 0 40 5
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,16,0,0,5,40],[1,0,0,0,0,1,0,0,0,0,4,9,0,0,30,37],[37,0,0,0,7,10,0,0,0,0,1,0,0,0,0,1],[10,20,0,0,34,31,0,0,0,0,36,40,0,0,24,5] >;

C5×Q82D5 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_2D_5
% in TeX

G:=Group("C5xQ8:2D5");
// GroupNames label

G:=SmallGroup(400,188);
// by ID

G=gap.SmallGroup(400,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,404,194,11525]);
// 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,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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