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## G = C5×C40⋊C2order 400 = 24·52

### Direct product of C5 and C40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C5×C40⋊C2
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×D20 — C5×C40⋊C2
 Lower central C5 — C10 — C20 — C5×C40⋊C2
 Upper central C1 — C10 — C20 — C40

Generators and relations for C5×C40⋊C2
G = < a,b,c | a5=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Smallest permutation representation of C5×C40⋊C2
On 80 points
Generators in S80
(1 9 17 25 33)(2 10 18 26 34)(3 11 19 27 35)(4 12 20 28 36)(5 13 21 29 37)(6 14 22 30 38)(7 15 23 31 39)(8 16 24 32 40)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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 77)(2 56)(3 75)(4 54)(5 73)(6 52)(7 71)(8 50)(9 69)(10 48)(11 67)(12 46)(13 65)(14 44)(15 63)(16 42)(17 61)(18 80)(19 59)(20 78)(21 57)(22 76)(23 55)(24 74)(25 53)(26 72)(27 51)(28 70)(29 49)(30 68)(31 47)(32 66)(33 45)(34 64)(35 43)(36 62)(37 41)(38 60)(39 79)(40 58)

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

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

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

115 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 ··· 20X 20Y 20Z 20AA 20AB 40A ··· 40AV 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 20 1 1 1 1 2 ··· 2 2 2 1 1 1 1 2 ··· 2 20 20 20 20 2 ··· 2 20 20 20 20 2 ··· 2

115 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D5 SD16 D10 D20 C5×D4 C5×D5 C40⋊C2 C5×SD16 D5×C10 C5×D20 C5×C40⋊C2 kernel C5×C40⋊C2 C5×C40 C5×Dic10 C5×D20 C40⋊C2 C40 Dic10 D20 C5×C10 C40 C52 C20 C10 C10 C8 C5 C5 C4 C2 C1 # reps 1 1 1 1 4 4 4 4 1 2 2 2 4 4 8 8 8 8 16 32

Matrix representation of C5×C40⋊C2 in GL2(𝔽41) generated by

 16 0 0 16
,
 29 0 0 24
,
 0 32 9 0
G:=sub<GL(2,GF(41))| [16,0,0,16],[29,0,0,24],[0,9,32,0] >;

C5×C40⋊C2 in GAP, Magma, Sage, TeX

C_5\times C_{40}\rtimes C_2
% in TeX

G:=Group("C5xC40:C2");
// GroupNames label

G:=SmallGroup(400,78);
// by ID

G=gap.SmallGroup(400,78);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,265,127,1443,69,11525]);
// Polycyclic

G:=Group<a,b,c|a^5=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations

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