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## G = C2×C22.F5order 160 = 25·5

### Direct product of C2 and C22.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C22.F5
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C2×C22.F5
 Lower central C5 — C10 — C2×C22.F5
 Upper central C1 — C22 — C23

Generators and relations for C2×C22.F5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 164 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C5⋊C8, C22.F5, C22×Dic5, C2×C22.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C22.F5, C22×F5, C2×C22.F5

Character table of C2×C22.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 10D 10E 10F 10G size 1 1 1 1 2 2 5 5 5 5 10 10 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 2i -2i -2i 2i 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 -2 complex lifted from M4(2) ρ18 2 -2 2 -2 0 0 -2i -2i 2i 2i 0 0 2 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 -2 complex lifted from M4(2) ρ19 2 -2 2 -2 0 0 2i 2i -2i -2i 0 0 2 0 0 0 0 0 0 0 0 0 -2 0 0 2 0 -2 complex lifted from M4(2) ρ20 2 2 -2 -2 0 0 -2i 2i 2i -2i 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 -2 0 -2 complex lifted from M4(2) ρ21 4 4 4 4 -4 -4 0 0 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 -4 4 0 0 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 4 -4 0 0 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 ρ24 4 4 4 4 4 4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 √5 -1 -√5 √5 1 -√5 1 symplectic lifted from C22.F5, Schur index 2 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -√5 1 -√5 √5 -1 √5 1 symplectic lifted from C22.F5, Schur index 2 ρ27 4 -4 4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 √5 1 √5 -√5 -1 -√5 1 symplectic lifted from C22.F5, Schur index 2 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -√5 -1 √5 -√5 1 √5 1 symplectic lifted from C22.F5, Schur index 2

Smallest permutation representation of C2×C22.F5
On 80 points
Generators in S80
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 73)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(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)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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( (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,65),(7,66),(8,67),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,73),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(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×C22.F5 is a maximal subgroup of
C22⋊C4.F5  C22.F5⋊C4  Dic5.C42  D10⋊M4(2)  Dic5⋊M4(2)  C20⋊C8⋊C2  D109M4(2)  C208M4(2)  C5⋊C87D4  C202M4(2)  (C2×D4).7F5  (C2×D4).9F5  C24.4F5  Dic5.C24
C2×C22.F5 is a maximal quotient of
C20.34M4(2)  Dic5.13M4(2)  C208M4(2)  C20.30M4(2)  C5⋊C87D4  C202M4(2)  C20.6M4(2)  C24.4F5

Matrix representation of C2×C22.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
,
 1 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 40 0 0 0 0 0 0 40
,
 40 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 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 33 7 0 0 0 0 0 0 34 6 0 0 0 0 34 0
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 19 0 0 0

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],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,6,0],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C22.F5 in GAP, Magma, Sage, TeX

C_2\times C_2^2.F_5
% in TeX

G:=Group("C2xC2^2.F5");
// GroupNames label

G:=SmallGroup(160,211);
// by ID

G=gap.SmallGroup(160,211);
# by ID

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

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

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