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## G = F5×C2×C10order 400 = 24·52

### Direct product of C2×C10 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C2×C10
 Chief series C1 — C5 — D5 — C5×D5 — C5×F5 — C10×F5 — F5×C2×C10
 Lower central C5 — F5×C2×C10
 Upper central C1 — C2×C10

Generators and relations for F5×C2×C10
G = < a,b,c,d | a2=b10=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 304 in 113 conjugacy classes, 64 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C22, C22 [×6], C5 [×2], C5, C2×C4 [×6], C23, D5, D5 [×3], C10 [×6], C10 [×7], C22×C4, C20 [×4], F5 [×4], D10 [×6], C2×C10 [×2], C2×C10 [×7], C52, C2×C20 [×6], C2×F5 [×6], C22×D5, C22×C10, C5×D5, C5×D5 [×3], C5×C10 [×3], C22×C20, C22×F5, C5×F5 [×4], D5×C10 [×6], C102, C10×F5 [×6], D5×C2×C10, F5×C2×C10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], C23, C10 [×7], C22×C4, C20 [×4], F5, C2×C10 [×7], C2×C20 [×6], C2×F5 [×3], C22×C10, C22×C20, C22×F5, C5×F5, C10×F5 [×3], F5×C2×C10

Smallest permutation representation of F5×C2×C10
On 80 points
Generators in S80
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 3 5 7 9)(2 4 6 8 10)(11 17 13 19 15)(12 18 14 20 16)(21 29 27 25 23)(22 30 28 26 24)(31 33 35 37 39)(32 34 36 38 40)(41 49 47 45 43)(42 50 48 46 44)(51 55 59 53 57)(52 56 60 54 58)(61 67 63 69 65)(62 68 64 70 66)(71 75 79 73 77)(72 76 80 74 78)
(1 16 25 73)(2 17 26 74)(3 18 27 75)(4 19 28 76)(5 20 29 77)(6 11 30 78)(7 12 21 79)(8 13 22 80)(9 14 23 71)(10 15 24 72)(31 63 48 56)(32 64 49 57)(33 65 50 58)(34 66 41 59)(35 67 42 60)(36 68 43 51)(37 69 44 52)(38 70 45 53)(39 61 46 54)(40 62 47 55)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5A 5B 5C 5D 5E ··· 5I 10A ··· 10L 10M ··· 10AA 10AB ··· 10AQ 20A ··· 20AF order 1 2 2 2 2 2 2 2 4 ··· 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 5 ··· 5 1 1 1 1 4 ··· 4 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 F5 C2×F5 C5×F5 C10×F5 kernel F5×C2×C10 C10×F5 D5×C2×C10 D5×C10 C102 C22×F5 C2×F5 C22×D5 D10 C2×C10 C2×C10 C10 C22 C2 # reps 1 6 1 6 2 4 24 4 24 8 1 3 4 12

Matrix representation of F5×C2×C10 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 31 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 16 0 0 0 0 17 18 0 0 0 10 0 10 0 0 6 0 0 37
,
 32 0 0 0 0 0 1 0 24 0 0 0 0 1 40 0 0 40 40 0 0 0 0 40 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[31,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,16,17,10,6,0,0,18,0,0,0,0,0,10,0,0,0,0,0,37],[32,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,24,1,40,40,0,0,40,0,0] >;

F5×C2×C10 in GAP, Magma, Sage, TeX

F_5\times C_2\times C_{10}
% in TeX

G:=Group("F5xC2xC10");
// GroupNames label

G:=SmallGroup(400,214);
// by ID

G=gap.SmallGroup(400,214);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,5765,317]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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