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## G = C2×2- 1+4⋊C5order 320 = 26·5

### Direct product of C2 and 2- 1+4⋊C5

Aliases: C2×2- 1+4⋊C5, 2- 1+4⋊C10, (C2×2- 1+4)⋊C5, C22.1(C24⋊C5), C2.2(C2×C24⋊C5), SmallGroup(320,1585)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2- 1+4 — C2×2- 1+4⋊C5
 Chief series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5 — C2×2- 1+4⋊C5
 Lower central 2- 1+4 — C2×2- 1+4⋊C5
 Upper central C1 — C22

Generators and relations for C2×2- 1+4⋊C5
G = < a,b,c,d,e,f | a2=b4=c2=f5=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2cde, cd=dc, ce=ec, fcf-1=bc, ede-1=b2d, fdf-1=bcd, fef-1=de >

Subgroups: 499 in 92 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C10, C22×Q8, C2×C4○D4, 2- 1+4, 2- 1+4, C2×2- 1+4, 2- 1+4⋊C5, C2×2- 1+4⋊C5
Quotients: C1, C2, C5, C10, C24⋊C5, 2- 1+4⋊C5, C2×C24⋊C5, C2×2- 1+4⋊C5

Character table of C2×2- 1+4⋊C5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L size 1 1 1 1 10 10 10 10 10 10 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 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 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 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 ζ5 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ54 linear of order 5 ρ4 1 1 1 1 1 1 1 1 1 1 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 ζ53 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ52 linear of order 5 ρ5 1 -1 1 -1 -1 1 1 1 -1 -1 ζ53 ζ54 ζ5 ζ52 ζ53 ζ54 ζ52 ζ5 -ζ5 -ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ54 -ζ54 linear of order 10 ρ6 1 -1 1 -1 -1 1 1 1 -1 -1 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 -ζ54 -ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ5 -ζ5 linear of order 10 ρ7 1 -1 1 -1 -1 1 1 1 -1 -1 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 -ζ52 -ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ53 -ζ53 linear of order 10 ρ8 1 1 1 1 1 1 1 1 1 1 ζ5 ζ53 ζ52 ζ54 ζ5 ζ53 ζ54 ζ52 ζ52 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ53 linear of order 5 ρ9 1 -1 1 -1 -1 1 1 1 -1 -1 ζ54 ζ52 ζ53 ζ5 ζ54 ζ52 ζ5 ζ53 -ζ53 -ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ52 -ζ52 linear of order 10 ρ10 1 1 1 1 1 1 1 1 1 1 ζ52 ζ5 ζ54 ζ53 ζ52 ζ5 ζ53 ζ54 ζ54 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ5 linear of order 5 ρ11 4 -4 -4 4 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 1 -1 1 -1 1 -1 1 -1 symplectic lifted from 2- 1+4⋊C5, Schur index 2 ρ12 4 4 -4 -4 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 symplectic lifted from 2- 1+4⋊C5, Schur index 2 ρ13 4 -4 -4 4 0 0 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ52 ζ5 ζ53 ζ54 ζ54 -ζ54 ζ53 -ζ53 ζ52 -ζ52 ζ5 -ζ5 complex lifted from 2- 1+4⋊C5 ρ14 4 4 -4 -4 0 0 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ52 ζ5 ζ53 ζ54 -ζ54 ζ54 -ζ53 ζ53 -ζ52 ζ52 -ζ5 ζ5 complex lifted from 2- 1+4⋊C5 ρ15 4 4 -4 -4 0 0 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 ζ5 ζ53 ζ54 ζ52 -ζ52 ζ52 -ζ54 ζ54 -ζ5 ζ5 -ζ53 ζ53 complex lifted from 2- 1+4⋊C5 ρ16 4 -4 -4 4 0 0 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 ζ5 ζ53 ζ54 ζ52 ζ52 -ζ52 ζ54 -ζ54 ζ5 -ζ5 ζ53 -ζ53 complex lifted from 2- 1+4⋊C5 ρ17 4 4 -4 -4 0 0 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 ζ53 ζ54 ζ52 ζ5 -ζ5 ζ5 -ζ52 ζ52 -ζ53 ζ53 -ζ54 ζ54 complex lifted from 2- 1+4⋊C5 ρ18 4 -4 -4 4 0 0 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 ζ53 ζ54 ζ52 ζ5 ζ5 -ζ5 ζ52 -ζ52 ζ53 -ζ53 ζ54 -ζ54 complex lifted from 2- 1+4⋊C5 ρ19 4 -4 -4 4 0 0 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ54 ζ52 ζ5 ζ53 ζ53 -ζ53 ζ5 -ζ5 ζ54 -ζ54 ζ52 -ζ52 complex lifted from 2- 1+4⋊C5 ρ20 4 4 -4 -4 0 0 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ54 ζ52 ζ5 ζ53 -ζ53 ζ53 -ζ5 ζ5 -ζ54 ζ54 -ζ52 ζ52 complex lifted from 2- 1+4⋊C5 ρ21 5 -5 5 -5 -1 1 -3 1 -1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ22 5 5 5 5 1 1 -3 1 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ23 5 -5 5 -5 3 -3 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ24 5 5 5 5 -3 -3 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ25 5 -5 5 -5 -1 1 1 -3 3 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ26 5 5 5 5 1 1 1 -3 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5

Smallest permutation representation of C2×2- 1+4⋊C5
On 64 points
Generators in S64
(1 3)(2 4)(5 22)(6 23)(7 24)(8 20)(9 21)(10 44)(11 40)(12 41)(13 42)(14 43)(15 50)(16 51)(17 52)(18 53)(19 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 61)(31 62)(32 63)(33 64)(34 60)(35 46)(36 47)(37 48)(38 49)(39 45)
(1 55 4 53)(2 18 3 25)(5 29 46 17)(6 19 47 26)(7 9 48 45)(8 34 49 44)(10 20 60 38)(11 57 61 50)(12 13 62 63)(14 58 64 51)(15 40 27 30)(16 43 28 33)(21 37 39 24)(22 59 35 52)(23 54 36 56)(31 32 41 42)
(1 63)(2 42)(3 32)(4 13)(5 40)(6 16)(7 34)(8 9)(10 37)(11 22)(12 53)(14 56)(15 29)(17 27)(18 41)(19 33)(20 21)(23 51)(24 60)(25 31)(26 43)(28 47)(30 46)(35 61)(36 58)(38 39)(44 48)(45 49)(50 59)(52 57)(54 64)(55 62)
(1 20 4 38)(2 49 3 8)(5 26 46 19)(6 17 47 29)(7 41 48 31)(9 42 45 32)(10 55 60 53)(11 14 61 64)(12 37 62 24)(13 39 63 21)(15 16 27 28)(18 44 25 34)(22 56 35 54)(23 52 36 59)(30 33 40 43)(50 51 57 58)
(1 58 4 51)(2 16 3 28)(5 7 46 48)(6 32 47 42)(8 27 49 15)(9 17 45 29)(10 11 60 61)(12 56 62 54)(13 23 63 36)(14 55 64 53)(18 43 25 33)(19 41 26 31)(20 57 38 50)(21 52 39 59)(22 24 35 37)(30 44 40 34)
(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)

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

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

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

Matrix representation of C2×2- 1+4⋊C5 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 39 5 24 38 0 5 39 38 24 0 24 38 2 36 0 38 24 36 2
,
 1 0 0 0 0 0 2 20 7 29 0 21 39 12 34 0 7 29 39 21 0 12 34 20 2
,
 1 0 0 0 0 0 20 36 13 30 0 5 21 11 28 0 28 11 20 36 0 30 13 5 21
,
 1 0 0 0 0 0 12 34 20 2 0 7 29 39 21 0 20 2 29 7 0 39 21 34 12
,
 37 0 0 0 0 0 1 0 0 0 0 7 29 39 21 0 0 0 0 40 0 21 39 12 34

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],[1,0,0,0,0,0,39,5,24,38,0,5,39,38,24,0,24,38,2,36,0,38,24,36,2],[1,0,0,0,0,0,2,21,7,12,0,20,39,29,34,0,7,12,39,20,0,29,34,21,2],[1,0,0,0,0,0,20,5,28,30,0,36,21,11,13,0,13,11,20,5,0,30,28,36,21],[1,0,0,0,0,0,12,7,20,39,0,34,29,2,21,0,20,39,29,34,0,2,21,7,12],[37,0,0,0,0,0,1,7,0,21,0,0,29,0,39,0,0,39,0,12,0,0,21,40,34] >;

C2×2- 1+4⋊C5 in GAP, Magma, Sage, TeX

C_2\times 2_-^{1+4}\rtimes C_5
% in TeX

G:=Group("C2xES-(2,2):C5");
// GroupNames label

G:=SmallGroup(320,1585);
// by ID

G=gap.SmallGroup(320,1585);
# by ID

G:=PCGroup([7,-2,-5,-2,2,2,2,-2,849,1270,521,248,1936,718,375,172,3162,1027]);
// Polycyclic

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

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