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## G = (C2×D4)⋊F5order 320 = 26·5

### 2nd semidirect product of C2×D4 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×D4)⋊F5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D10.D4 — (C2×D4)⋊F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — (C2×D4)⋊F5
 Upper central C1 — C2 — C22 — C2×C4 — C2×D4

Generators and relations for (C2×D4)⋊F5
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 586 in 86 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×D4, Dic5, C20, F5, D10, C2×C10, C2×C10, C23⋊C4, C41D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C22×D5, C22×C10, C42⋊C4, C4×Dic5, C22⋊F5, C2×D20, C2×C5⋊D4, D4×C10, D10.D4, C20⋊D4, (C2×D4)⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊C4, C22⋊F5, C23⋊F5, (C2×D4)⋊F5

Character table of (C2×D4)⋊F5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 5 10A 10B 10C 10D 10E 10F 10G 20A 20B size 1 1 2 8 20 20 4 20 20 40 40 40 40 4 4 4 4 8 8 8 8 8 8 ρ1 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 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 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 linear of order 2 ρ5 1 1 1 1 -1 -1 1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 1 i i -i -i 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 1 1 -i -i i i 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ9 2 2 2 0 2 -2 -2 0 0 0 0 0 0 2 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 0 -2 2 -2 0 0 0 0 0 0 2 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ11 4 -4 0 0 0 0 0 -2 2 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ12 4 -4 0 0 0 0 0 2 -2 0 0 0 0 4 0 -4 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ13 4 4 4 4 0 0 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 0 0 0 0 0 0 4 -4 4 -4 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ15 4 4 4 -4 0 0 4 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ16 4 4 4 0 0 0 -4 0 0 0 0 0 0 -1 -1 -1 -1 √5 √5 -√5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ17 4 4 4 0 0 0 -4 0 0 0 0 0 0 -1 -1 -1 -1 -√5 -√5 √5 √5 1 1 orthogonal lifted from C22⋊F5 ρ18 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 √5 -√5 complex lifted from C23⋊F5 ρ19 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 √5 -√5 complex lifted from C23⋊F5 ρ20 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 -√5 √5 complex lifted from C23⋊F5 ρ21 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 -√5 √5 complex lifted from C23⋊F5 ρ22 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 -2√5 2 2√5 0 0 0 0 0 0 orthogonal faithful ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 2√5 2 -2√5 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of (C2×D4)⋊F5
On 40 points
Generators in S40
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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)
(1 33 11 23)(2 35 15 21)(3 32 14 24)(4 34 13 22)(5 31 12 25)(6 38 16 28)(7 40 20 26)(8 37 19 29)(9 39 18 27)(10 36 17 30)

G:=sub<Sym(40)| (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,6)(2,7)(3,8)(4,9)(5,10)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,33,11,23)(2,35,15,21)(3,32,14,24)(4,34,13,22)(5,31,12,25)(6,38,16,28)(7,40,20,26)(8,37,19,29)(9,39,18,27)(10,36,17,30)>;

G:=Group( (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,6)(2,7)(3,8)(4,9)(5,10)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (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), (1,33,11,23)(2,35,15,21)(3,32,14,24)(4,34,13,22)(5,31,12,25)(6,38,16,28)(7,40,20,26)(8,37,19,29)(9,39,18,27)(10,36,17,30) );

G=PermutationGroup([[(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,11,6,16),(2,12,7,17),(3,13,8,18),(4,14,9,19),(5,15,10,20),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(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)], [(1,33,11,23),(2,35,15,21),(3,32,14,24),(4,34,13,22),(5,31,12,25),(6,38,16,28),(7,40,20,26),(8,37,19,29),(9,39,18,27),(10,36,17,30)]])

Matrix representation of (C2×D4)⋊F5 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 40 0 0 0 0 0 39 29 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 25 0 0 0 0 0 0 36 1 0 0 0 0 0 0 5 29 40 5 0 0 0 0 31 16 16 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 40 0 0 0 0 0 0 36 11 1 36 0 0 0 0 9 0 0 40
,
 40 40 0 0 0 0 0 0 36 35 0 0 0 0 0 0 39 39 6 40 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 3 1 1 0 0 0 0 21 18 6 40 0 0 0 0 25 25 20 0 0 0 0 0 0 10 3 3 0 0 0 0 0 0 0 0 40 0 16 0 0 0 0 0 0 30 39 5 0 0 0 0 0 1 1 0 0 0 0 0 0 17 37 11

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,39,0,0,0,0,0,1,0,29,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,36,5,31,0,0,0,0,25,1,29,16,0,0,0,0,0,0,40,16,0,0,0,0,0,0,5,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,5,36,9,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,40],[40,36,39,0,0,0,0,0,40,35,39,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,21,25,0,0,0,0,0,3,18,25,10,0,0,0,0,1,6,20,3,0,0,0,0,1,40,0,3,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,30,1,17,0,0,0,0,16,39,1,37,0,0,0,0,0,5,0,11] >;

(C2×D4)⋊F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes F_5
% in TeX

G:=Group("(C2xD4):F5");
// GroupNames label

G:=SmallGroup(320,260);
// by ID

G=gap.SmallGroup(320,260);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,297,136,1684,6278,3156]);
// Polycyclic

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

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