Copied to
clipboard

G = C2×C8⋊F5order 320 = 26·5

Direct product of C2 and C8⋊F5

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

Aliases: C2×C8⋊F5, C20.30C42, D10.12C42, D10.4M4(2), Dic5.10C42, C89(C2×F5), (C2×C8)⋊8F5, (C2×C40)⋊9C4, D5⋊C86C4, C4010(C2×C4), (C8×D5)⋊12C4, (C4×F5).5C4, C4.23(C4×F5), C101(C8⋊C4), D52(C8⋊C4), (C22×F5).3C4, C4.49(C22×F5), C22.19(C4×F5), (C2×C10).17C42, C10.12(C2×C42), C20.89(C22×C4), D5⋊C8.17C22, (C4×D5).86C23, (C8×D5).65C22, D5.1(C2×M4(2)), (C4×F5).16C22, D10.32(C22×C4), Dic5.31(C22×C4), (C2×C5⋊C8)⋊6C4, C5⋊C85(C2×C4), C52(C2×C8⋊C4), C2.13(C2×C4×F5), (D5×C2×C8).33C2, (C2×C52C8)⋊23C4, C52C837(C2×C4), (C2×C4×F5).12C2, (C2×F5).6(C2×C4), (C2×D5⋊C8).10C2, (C4×D5).95(C2×C4), (C2×C4).166(C2×F5), (C2×C20).175(C2×C4), (C2×C4×D5).413C22, (C22×D5).87(C2×C4), (C2×Dic5).125(C2×C4), SmallGroup(320,1055)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C8⋊F5
C1C5C10D10C4×D5C4×F5C2×C4×F5 — C2×C8⋊F5
C5C10 — C2×C8⋊F5
C1C2×C4C2×C8

Generators and relations for C2×C8⋊F5
 G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >

Subgroups: 442 in 146 conjugacy classes, 76 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×13], C23, D5 [×4], C10, C10 [×2], C42 [×4], C2×C8, C2×C8 [×11], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C8⋊C4 [×4], C2×C42, C22×C8 [×2], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C2×C8⋊C4, C8×D5 [×4], C2×C52C8, C2×C40, D5⋊C8 [×4], C4×F5 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×F5 [×2], C8⋊F5 [×4], D5×C2×C8, C2×D5⋊C8, C2×C4×F5, C2×C8⋊F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C2×F5 [×3], C2×C8⋊C4, C4×F5 [×2], C22×F5, C8⋊F5 [×2], C2×C4×F5, C2×C8⋊F5

Smallest permutation representation of C2×C8⋊F5
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 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)
(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 59 51 42 65)(2 60 52 43 66)(3 61 53 44 67)(4 62 54 45 68)(5 63 55 46 69)(6 64 56 47 70)(7 57 49 48 71)(8 58 50 41 72)(9 17 33 25 74)(10 18 34 26 75)(11 19 35 27 76)(12 20 36 28 77)(13 21 37 29 78)(14 22 38 30 79)(15 23 39 31 80)(16 24 40 32 73)
(1 17 5 21)(2 22 6 18)(3 19 7 23)(4 24 8 20)(9 46 37 51)(10 43 38 56)(11 48 39 53)(12 45 40 50)(13 42 33 55)(14 47 34 52)(15 44 35 49)(16 41 36 54)(25 69 78 59)(26 66 79 64)(27 71 80 61)(28 68 73 58)(29 65 74 63)(30 70 75 60)(31 67 76 57)(32 72 77 62)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4P 5 8A8B8C8D8E···8P10A10B10C20A20B20C20D40A···40H
order12222222444444444···4588888···81010102020202040···40
size111155551111555510···104222210···1044444444···4

56 irreducible representations

dim1111111111112444444
type++++++++
imageC1C2C2C2C2C4C4C4C4C4C4C4M4(2)F5C2×F5C2×F5C4×F5C4×F5C8⋊F5
kernelC2×C8⋊F5C8⋊F5D5×C2×C8C2×D5⋊C8C2×C4×F5C8×D5C2×C52C8C2×C40D5⋊C8C4×F5C2×C5⋊C8C22×F5D10C2×C8C8C2×C4C4C22C2
# reps1411142244448121228

Matrix representation of C2×C8⋊F5 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
1390000
37400000
0032000
0003200
0000320
0000032
,
100000
010000
0040100
0040010
0040001
0040000
,
40200000
010000
0010400
0000401
0001400
0000400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,39,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40,0,0,0,1,0,0] >;

C2×C8⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes F_5
% in TeX

G:=Group("C2xC8:F5");
// GroupNames label

G:=SmallGroup(320,1055);
// by ID

G=gap.SmallGroup(320,1055);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,102,6278,1595]);
// 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,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽