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G = C2xC8xF5order 320 = 26·5

Direct product of C2xC8 and F5

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

Aliases: C2xC8xF5, C20.29C42, D10.11C42, Dic5.9C42, (C2xC40):8C4, C40:9(C2xC4), C10:1(C4xC8), D5:C8:7C4, D5:2(C4xC8), (C8xD5):11C4, (C4xF5).7C4, C4.22(C4xF5), D5.(C22xC8), D10.7(C2xC8), (C22xF5).5C4, C4.48(C22xF5), C22.18(C4xF5), (C2xC10).16C42, C10.11(C2xC42), C20.88(C22xC4), D5:C8.21C22, (C8xD5).64C22, (C4xD5).85C23, (C4xF5).20C22, D10.31(C22xC4), Dic5.30(C22xC4), C5:2(C2xC4xC8), (C2xC5:C8):9C4, C5:C8:8(C2xC4), C2.3(C2xC4xF5), (D5xC2xC8).32C2, (C2xC5:2C8):22C4, C5:2C8:36(C2xC4), (C2xC4xF5).14C2, (C2xF5).9(C2xC4), (C2xD5:C8).12C2, (C4xD5).94(C2xC4), (C2xC4).165(C2xF5), (C2xC20).174(C2xC4), (C2xC4xD5).412C22, (C22xD5).86(C2xC4), (C2xDic5).124(C2xC4), SmallGroup(320,1054)

Series: Derived Chief Lower central Upper central

C1C5 — C2xC8xF5
C1C5C10D10C4xD5C4xF5C2xC4xF5 — C2xC8xF5
C5 — C2xC8xF5
C1C2xC8

Generators and relations for C2xC8xF5
 G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 442 in 162 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, C23, D5, C10, C10, C42, C2xC8, C2xC8, C22xC4, Dic5, C20, F5, D10, D10, C2xC10, C4xC8, C2xC42, C22xC8, C5:2C8, C40, C5:C8, C4xD5, C2xDic5, C2xC20, C2xF5, C22xD5, C2xC4xC8, C8xD5, C2xC5:2C8, C2xC40, D5:C8, C4xF5, C2xC5:C8, C2xC4xD5, C22xF5, C8xF5, D5xC2xC8, C2xD5:C8, C2xC4xF5, C2xC8xF5
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C42, C2xC8, C22xC4, F5, C4xC8, C2xC42, C22xC8, C2xF5, C2xC4xC8, C4xF5, C22xF5, C8xF5, C2xC4xF5, C2xC8xF5

Smallest permutation representation of C2xC8xF5
On 80 points
Generators in S80
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)
(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 65 48 56 59)(2 66 41 49 60)(3 67 42 50 61)(4 68 43 51 62)(5 69 44 52 63)(6 70 45 53 64)(7 71 46 54 57)(8 72 47 55 58)(9 74 30 33 17)(10 75 31 34 18)(11 76 32 35 19)(12 77 25 36 20)(13 78 26 37 21)(14 79 27 38 22)(15 80 28 39 23)(16 73 29 40 24)
(1 5)(2 6)(3 7)(4 8)(9 78 33 26)(10 79 34 27)(11 80 35 28)(12 73 36 29)(13 74 37 30)(14 75 38 31)(15 76 39 32)(16 77 40 25)(17 21)(18 22)(19 23)(20 24)(41 64 49 70)(42 57 50 71)(43 58 51 72)(44 59 52 65)(45 60 53 66)(46 61 54 67)(47 62 55 68)(48 63 56 69)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4X 5 8A···8H8I···8AF10A10B10C20A20B20C20D40A···40H
order1222222244444···458···88···81010102020202040···40
size1111555511115···541···15···544444444···4

80 irreducible representations

dim1111111111111444444
type++++++++
imageC1C2C2C2C2C4C4C4C4C4C4C4C8F5C2xF5C2xF5C4xF5C4xF5C8xF5
kernelC2xC8xF5C8xF5D5xC2xC8C2xD5:C8C2xC4xF5C8xD5C2xC5:2C8C2xC40D5:C8C4xF5C2xC5:C8C22xF5C2xF5C2xC8C8C2xC4C4C22C2
# reps14111422444432121228

Matrix representation of C2xC8xF5 in GL5(F41)

400000
040000
004000
000400
000040
,
90000
027000
002700
000270
000027
,
10000
040404040
01000
00100
00010
,
90000
040000
000040
004000
01111

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,40,0,0,0],[9,0,0,0,0,0,40,0,0,1,0,0,0,40,1,0,0,0,0,1,0,0,40,0,1] >;

C2xC8xF5 in GAP, Magma, Sage, TeX

C_2\times C_8\times F_5
% in TeX

G:=Group("C2xC8xF5");
// GroupNames label

G:=SmallGroup(320,1054);
// by ID

G=gap.SmallGroup(320,1054);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=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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