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## G = C2×C5⋊C8order 80 = 24·5

### Direct product of C2 and C5⋊C8

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

 Derived series C1 — C5 — C2×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8
 Lower central C5 — C2×C5⋊C8
 Upper central C1 — C22

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

Character table of C2×C5⋊C8

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C size 1 1 1 1 5 5 5 5 4 5 5 5 5 5 5 5 5 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 i -i -i i i -i -i i 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 1 -i i i -i -i i i -i 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 1 1 -i -i -i i i i i -i -1 -1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 -1 1 1 i i i -i -i -i -i i -1 -1 1 linear of order 4 ρ9 1 -1 1 -1 -i i -i i 1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 1 -1 -1 linear of order 8 ρ10 1 -1 1 -1 i -i i -i 1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 1 -1 -1 linear of order 8 ρ11 1 -1 1 -1 -i i -i i 1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 1 -1 -1 linear of order 8 ρ12 1 -1 1 -1 i -i i -i 1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 1 -1 -1 linear of order 8 ρ13 1 -1 -1 1 -i -i i i 1 ζ87 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 -1 1 -1 linear of order 8 ρ14 1 -1 -1 1 i i -i -i 1 ζ8 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 -1 1 -1 linear of order 8 ρ15 1 -1 -1 1 i i -i -i 1 ζ85 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 -1 1 -1 linear of order 8 ρ16 1 -1 -1 1 -i -i i i 1 ζ83 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 -1 1 -1 linear of order 8 ρ17 4 4 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ18 4 4 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 orthogonal lifted from C2×F5 ρ19 4 -4 -4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 symplectic lifted from C5⋊C8, Schur index 2 ρ20 4 -4 4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 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)

 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 0 0 40 0 1 0 0 40 0 0 1 0 40 0 0 0 1 40
,
 40 0 0 0 0 0 9 0 12 31 0 21 31 1 40 0 10 40 1 11 0 10 11 32 0

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

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