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G = C2×C5⋊C8order 80 = 24·5

Direct product of C2 and C5⋊C8

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

Aliases: C2×C5⋊C8, C10⋊C8, C22.2F5, Dic5.2C4, Dic5.6C22, C52(C2×C8), C2.3(C2×F5), (C2×C10).1C4, C10.5(C2×C4), (C2×Dic5).4C2, SmallGroup(80,32)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C5⋊C8
C1C5C10Dic5C5⋊C8 — C2×C5⋊C8
C5 — C2×C5⋊C8
C1C22

Generators and relations for C2×C5⋊C8
 G = < a,b,c | a2=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

5C4
5C4
5C8
5C2×C4
5C8
5C2×C8

Character table of C2×C5⋊C8

 class 12A2B2C4A4B4C4D58A8B8C8D8E8F8G8H10A10B10C
 size 11115555455555555444
ρ111111111111111111111    trivial
ρ211-1-1-111-111-1-1-1-1111-1-11    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ411-1-1-111-11-11111-1-1-1-1-11    linear of order 2
ρ51111-1-1-1-11i-i-iii-i-ii111    linear of order 4
ρ61111-1-1-1-11-iii-i-iii-i111    linear of order 4
ρ711-1-11-1-111-i-i-iiiii-i-1-11    linear of order 4
ρ811-1-11-1-111iii-i-i-i-ii-1-11    linear of order 4
ρ91-11-1-ii-ii1ζ85ζ83ζ87ζ8ζ85ζ83ζ87ζ81-1-1    linear of order 8
ρ101-11-1i-ii-i1ζ87ζ8ζ85ζ83ζ87ζ8ζ85ζ831-1-1    linear of order 8
ρ111-11-1-ii-ii1ζ8ζ87ζ83ζ85ζ8ζ87ζ83ζ851-1-1    linear of order 8
ρ121-11-1i-ii-i1ζ83ζ85ζ8ζ87ζ83ζ85ζ8ζ871-1-1    linear of order 8
ρ131-1-11-i-iii1ζ87ζ85ζ8ζ87ζ83ζ8ζ85ζ83-11-1    linear of order 8
ρ141-1-11ii-i-i1ζ8ζ83ζ87ζ8ζ85ζ87ζ83ζ85-11-1    linear of order 8
ρ151-1-11ii-i-i1ζ85ζ87ζ83ζ85ζ8ζ83ζ87ζ8-11-1    linear of order 8
ρ161-1-11-i-iii1ζ83ζ8ζ85ζ83ζ87ζ85ζ8ζ87-11-1    linear of order 8
ρ1744440000-100000000-1-1-1    orthogonal lifted from F5
ρ1844-4-40000-10000000011-1    orthogonal lifted from C2×F5
ρ194-4-440000-1000000001-11    symplectic lifted from C5⋊C8, Schur index 2
ρ204-44-40000-100000000-111    symplectic lifted from C5⋊C8, Schur index 2

Smallest permutation representation of C2×C5⋊C8
Regular action on 80 points
Generators in S80
(1 72)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(1 19 75 35 57)(2 36 20 58 76)(3 59 37 77 21)(4 78 60 22 38)(5 23 79 39 61)(6 40 24 62 80)(7 63 33 73 17)(8 74 64 18 34)(9 45 71 29 50)(10 30 46 51 72)(11 52 31 65 47)(12 66 53 48 32)(13 41 67 25 54)(14 26 42 55 68)(15 56 27 69 43)(16 70 49 44 28)
(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)

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

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

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

C2×C5⋊C8 is a maximal subgroup of
C20⋊C8  C10.C42  D10⋊C8  Dic5⋊C8  C23.2F5  D4.F5  D10.2F5  C22.2S5
C2×C5⋊C8 is a maximal quotient of
C20.C8  C20⋊C8  C23.2F5  D10.2F5

Matrix representation of C2×C5⋊C8 in GL5(𝔽41)

400000
040000
004000
000400
000040
,
10000
000040
010040
001040
000140
,
400000
0901231
02131140
01040111
01011320

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[40,0,0,0,0,0,9,21,10,10,0,0,31,40,11,0,12,1,1,32,0,31,40,11,0] >;

C2×C5⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes C_8
% in TeX

G:=Group("C2xC5:C8");
// GroupNames label

G:=SmallGroup(80,32);
// by ID

G=gap.SmallGroup(80,32);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,42,804,414]);
// Polycyclic

G:=Group<a,b,c|a^2=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C2×C5⋊C8 in TeX
Character table of C2×C5⋊C8 in TeX

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