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G = C42xF5order 320 = 26·5

Direct product of C42 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C42xF5, C5:C43, C20:3C42, Dic5:4C42, (C4xC20):10C4, D5.(C2xC42), (C4xDic5):25C4, C10.4(C2xC42), (D5xC42).31C2, D10.26(C22xC4), C22.30(C22xF5), (C22xF5).21C22, (C22xD5).262C23, C2.2(C2xC4xF5), (C2xC4xF5).13C2, (C2xF5).8(C2xC4), (C4xD5).71(C2xC4), (C2xC4).160(C2xF5), (C2xC20).169(C2xC4), (C2xC4xD5).411C22, (C2xC10).23(C22xC4), (C2xDic5).171(C2xC4), SmallGroup(320,1023)

Series: Derived Chief Lower central Upper central

C1C5 — C42xF5
C1C5D5D10C22xD5C22xF5C2xC4xF5 — C42xF5
C5 — C42xF5
C1C42

Generators and relations for C42xF5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 714 in 258 conjugacy classes, 144 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C23, D5, C10, C42, C42, C22xC4, Dic5, C20, F5, D10, C2xC10, C2xC42, C4xD5, C2xDic5, C2xC20, C2xF5, C22xD5, C43, C4xDic5, C4xC20, C4xF5, C2xC4xD5, C22xF5, D5xC42, C2xC4xF5, C42xF5
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, F5, C2xC42, C2xF5, C43, C4xF5, C22xF5, C2xC4xF5, C42xF5

Smallest permutation representation of C42xF5
On 80 points
Generators in S80
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 61 16 76)(2 63 20 79)(3 65 19 77)(4 62 18 80)(5 64 17 78)(6 66 11 71)(7 68 15 74)(8 70 14 72)(9 67 13 75)(10 69 12 73)(21 51 36 46)(22 53 40 49)(23 55 39 47)(24 52 38 50)(25 54 37 48)(26 56 31 41)(27 58 35 44)(28 60 34 42)(29 57 33 45)(30 59 32 43)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4BD 5 10A10B10C20A···20L
order122222224···44···4510101020···20
size111155551···15···544444···4

80 irreducible representations

dim111111444
type+++++
imageC1C2C2C4C4C4F5C2xF5C4xF5
kernelC42xF5D5xC42C2xC4xF5C4xDic5C4xC20C4xF5C42C2xC4C4
# reps11662481312

Matrix representation of C42xF5 in GL6(F41)

900000
010000
0040000
0004000
0000400
0000040
,
100000
0320000
0040000
0004000
0000400
0000040
,
100000
010000
0040404040
001000
000100
000010
,
100000
010000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C42xF5 in GAP, Magma, Sage, TeX

C_4^2\times F_5
% in TeX

G:=Group("C4^2xF5");
// GroupNames label

G:=SmallGroup(320,1023);
// by ID

G=gap.SmallGroup(320,1023);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,100,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=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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