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G = F5xC2xC10order 400 = 24·52

Direct product of C2xC10 and F5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: F5xC2xC10, D10:3C20, C102:5C4, C10:(C2xC20), D5:(C2xC20), C5:(C22xC20), (C2xC10):2C20, (D5xC10):10C4, D5.(C22xC10), C52:4(C22xC4), D10.7(C2xC10), (C5xD5).2C23, (C22xD5).3C10, (D5xC10).23C22, (C5xC10):3(C2xC4), (C5xD5):4(C2xC4), (D5xC2xC10).6C2, SmallGroup(400,214)

Series: Derived Chief Lower central Upper central

C1C5 — F5xC2xC10
C1C5D5C5xD5C5xF5C10xF5 — F5xC2xC10
C5 — F5xC2xC10
C1C2xC10

Generators and relations for F5xC2xC10
 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, C2, C4, C22, C22, C5, C5, C2xC4, C23, D5, D5, C10, C10, C22xC4, C20, F5, D10, C2xC10, C2xC10, C52, C2xC20, C2xF5, C22xD5, C22xC10, C5xD5, C5xD5, C5xC10, C22xC20, C22xF5, C5xF5, D5xC10, C102, C10xF5, D5xC2xC10, F5xC2xC10
Quotients: C1, C2, C4, C22, C5, C2xC4, C23, C10, C22xC4, C20, F5, C2xC10, C2xC20, C2xF5, C22xC10, C22xC20, C22xF5, C5xF5, C10xF5, F5xC2xC10

Smallest permutation representation of F5xC2xC10
On 80 points
Generators in S80
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(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 75)(2 17 26 76)(3 18 27 77)(4 19 28 78)(5 20 29 79)(6 11 30 80)(7 12 21 71)(8 13 22 72)(9 14 23 73)(10 15 24 74)(31 61 46 56)(32 62 47 57)(33 63 48 58)(34 64 49 59)(35 65 50 60)(36 66 41 51)(37 67 42 52)(38 68 43 53)(39 69 44 54)(40 70 45 55)

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

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

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

100 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H5A5B5C5D5E···5I10A···10L10M···10AA10AB···10AQ20A···20AF
order122222224···455555···510···1010···1010···1020···20
size111155555···511114···41···14···45···55···5

100 irreducible representations

dim11111111114444
type+++++
imageC1C2C2C4C4C5C10C10C20C20F5C2xF5C5xF5C10xF5
kernelF5xC2xC10C10xF5D5xC2xC10D5xC10C102C22xF5C2xF5C22xD5D10C2xC10C2xC10C10C22C2
# reps16162424424813412

Matrix representation of F5xC2xC10 in GL5(F41)

400000
01000
00100
00010
00001
,
310000
04000
00400
00040
00004
,
10000
016000
0171800
0100100
060037
,
320000
010240
000140
0040400
000400

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] >;

F5xC2xC10 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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