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## G = C2×C4.F5order 160 = 25·5

### Direct product of C2 and C4.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4.F5
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C2×C4.F5
 Lower central C5 — C10 — C2×C4.F5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C4.F5
G = < a,b,c,d | a2=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 196 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C4.F5, C22×F5, C2×C4.F5

Character table of C2×C4.F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 10 10 2 2 5 5 5 5 4 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 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 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 -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 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ9 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -i -i i i -i -i i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ10 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -i i -i i -i i -i i 1 1 1 1 1 1 1 linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 i i -i -i i i -i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ12 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 i -i i -i i -i i -i 1 1 1 1 1 1 1 linear of order 4 ρ13 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 -i i -i -i i -i i i 1 -1 -1 -1 1 -1 1 linear of order 4 ρ14 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -i -i i -i i i -i i 1 -1 -1 1 -1 1 -1 linear of order 4 ρ15 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 i -i i i -i i -i -i 1 -1 -1 -1 1 -1 1 linear of order 4 ρ16 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 i i -i i -i -i i -i 1 -1 -1 1 -1 1 -1 linear of order 4 ρ17 2 -2 2 -2 0 0 0 0 -2i 2i 2i -2i 2 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 complex lifted from M4(2) ρ18 2 -2 -2 2 0 0 0 0 -2i 2i -2i 2i 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 2 -2 0 0 0 0 2i -2i -2i 2i 2 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 -2 2 0 0 0 0 2i -2i 2i -2i 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ21 4 4 -4 -4 0 0 -4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 1 -1 1 -1 orthogonal lifted from C2×F5 ρ22 4 4 -4 -4 0 0 4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -1 1 -1 1 orthogonal lifted from C2×F5 ρ23 4 4 4 4 0 0 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 4 4 0 0 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -√-5 -√-5 √-5 √-5 complex lifted from C4.F5 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 √-5 √-5 -√-5 -√-5 complex lifted from C4.F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 √-5 -√-5 -√-5 √-5 complex lifted from C4.F5 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 -√-5 √-5 √-5 -√-5 complex lifted from C4.F5

Smallest permutation representation of C2×C4.F5
On 80 points
Generators in S80
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 75)
(1 7 5 3)(2 4 6 8)(9 52 13 56)(10 49 14 53)(11 54 15 50)(12 51 16 55)(17 61 21 57)(18 58 22 62)(19 63 23 59)(20 60 24 64)(25 43 29 47)(26 48 30 44)(27 45 31 41)(28 42 32 46)(33 79 37 75)(34 76 38 80)(35 73 39 77)(36 78 40 74)(65 67 69 71)(66 72 70 68)
(1 57 35 75 19)(2 76 58 20 36)(3 21 77 37 59)(4 38 22 60 78)(5 61 39 79 23)(6 80 62 24 40)(7 17 73 33 63)(8 34 18 64 74)(9 50 29 71 45)(10 72 51 46 30)(11 47 65 31 52)(12 32 48 53 66)(13 54 25 67 41)(14 68 55 42 26)(15 43 69 27 56)(16 28 44 49 70)
(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)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (1,7,5,3)(2,4,6,8)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,43,29,47)(26,48,30,44)(27,45,31,41)(28,42,32,46)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(65,67,69,71)(66,72,70,68), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (1,7,5,3)(2,4,6,8)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,43,29,47)(26,48,30,44)(27,45,31,41)(28,42,32,46)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(65,67,69,71)(66,72,70,68), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,76),(42,77),(43,78),(44,79),(45,80),(46,73),(47,74),(48,75)], [(1,7,5,3),(2,4,6,8),(9,52,13,56),(10,49,14,53),(11,54,15,50),(12,51,16,55),(17,61,21,57),(18,58,22,62),(19,63,23,59),(20,60,24,64),(25,43,29,47),(26,48,30,44),(27,45,31,41),(28,42,32,46),(33,79,37,75),(34,76,38,80),(35,73,39,77),(36,78,40,74),(65,67,69,71),(66,72,70,68)], [(1,57,35,75,19),(2,76,58,20,36),(3,21,77,37,59),(4,38,22,60,78),(5,61,39,79,23),(6,80,62,24,40),(7,17,73,33,63),(8,34,18,64,74),(9,50,29,71,45),(10,72,51,46,30),(11,47,65,31,52),(12,32,48,53,66),(13,54,25,67,41),(14,68,55,42,26),(15,43,69,27,56),(16,28,44,49,70)], [(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×C4.F5 is a maximal subgroup of
C20.C42  C20.10C42  M4(2)⋊4F5  C203M4(2)  C42.14F5  C5⋊C8⋊D4  Dic5⋊M4(2)  D10.C42  D102M4(2)  C20⋊M4(2)  C4⋊C4.9F5  M4(2).1F5  D1010M4(2)  (C2×D4).8F5  (C2×Q8).5F5  D4⋊F5⋊C2  Dic5.22C24
C2×C4.F5 is a maximal quotient of
C42.12F5  C203M4(2)  C42.15F5  C5⋊C8⋊D4  C20⋊M4(2)  C20.M4(2)  D1010M4(2)  C20.30M4(2)

Matrix representation of C2×C4.F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 0 0 0 0 0 1 9 0 0 0 0 0 0 7 27 0 14 0 0 0 34 27 14 0 0 14 27 34 0 0 0 14 0 27 7
,
 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
,
 9 39 0 0 0 0 36 32 0 0 0 0 0 0 32 32 38 24 0 0 29 15 35 15 0 0 26 6 26 12 0 0 17 3 9 9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,1,0,0,0,0,0,9,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[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],[9,36,0,0,0,0,39,32,0,0,0,0,0,0,32,29,26,17,0,0,32,15,6,3,0,0,38,35,26,9,0,0,24,15,12,9] >;

C2×C4.F5 in GAP, Magma, Sage, TeX

C_2\times C_4.F_5
% in TeX

G:=Group("C2xC4.F5");
// GroupNames label

G:=SmallGroup(160,201);
// by ID

G=gap.SmallGroup(160,201);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,69,2309,599]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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