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G = (C2×D4)⋊F5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4)⋊2F5, (D4×C10)⋊2C4, (C4×Dic5)⋊5C4, C52(C42⋊C4), C20⋊D4.2C2, C2.6(C23⋊F5), D10.D42C2, (C22×D5).12D4, C10.15(C23⋊C4), (C2×D20).40C22, C22.19(C22⋊F5), (C2×C4).1(C2×F5), (C2×C20).11(C2×C4), (C2×C10).35(C22⋊C4), SmallGroup(320,260)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×D4)⋊F5
C1C5C10C2×C10C22×D5C2×D20D10.D4 — (C2×D4)⋊F5
C5C10C2×C10C2×C20 — (C2×D4)⋊F5
C1C2C22C2×C4C2×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 [×4], C4 [×5], C22, C22 [×7], C5, C2×C4, C2×C4 [×3], D4 [×6], C23 [×3], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C2×D4, C2×D4 [×3], Dic5 [×2], C20, F5 [×2], D10 [×4], C2×C10, C2×C10 [×3], C23⋊C4 [×2], C41D4, D20, C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4, C2×F5 [×2], C22×D5 [×2], C22×C10, C42⋊C4, C4×Dic5, C22⋊F5 [×2], C2×D20, C2×C5⋊D4 [×2], D4×C10, D10.D4 [×2], C20⋊D4, (C2×D4)⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊C4, C22⋊F5, C23⋊F5, (C2×D4)⋊F5

Character table of (C2×D4)⋊F5

 class 12A2B2C2D2E4A4B4C4D4E4F4G510A10B10C10D10E10F10G20A20B
 size 1128202042020404040404444888888
ρ111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111111    linear of order 2
ρ3111-1111-1-11-1-111111-1-1-1-111    linear of order 2
ρ4111-1111-1-1-111-11111-1-1-1-111    linear of order 2
ρ51111-1-11-1-1i-ii-i1111111111    linear of order 4
ρ61111-1-11-1-1-ii-ii1111111111    linear of order 4
ρ7111-1-1-1111ii-i-i1111-1-1-1-111    linear of order 4
ρ8111-1-1-1111-i-iii1111-1-1-1-111    linear of order 4
ρ922202-2-200000022220000-2-2    orthogonal lifted from D4
ρ102220-22-200000022220000-2-2    orthogonal lifted from D4
ρ114-400000-22000040-40000000    orthogonal lifted from C42⋊C4
ρ124-4000002-2000040-40000000    orthogonal lifted from C42⋊C4
ρ134444004000000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444-400000000004-44-4000000    orthogonal lifted from C23⋊C4
ρ15444-4004000000-1-1-1-11111-1-1    orthogonal lifted from C2×F5
ρ16444000-4000000-1-1-1-155-5-511    orthogonal lifted from C22⋊F5
ρ17444000-4000000-1-1-1-1-5-55511    orthogonal lifted from C22⋊F5
ρ1844-40000000000-11-1154+2ζ53+152+2ζ5+154+2ζ52+153+2ζ5+15-5    complex lifted from C23⋊F5
ρ1944-40000000000-11-1152+2ζ5+154+2ζ53+153+2ζ5+154+2ζ52+15-5    complex lifted from C23⋊F5
ρ2044-40000000000-11-1153+2ζ5+154+2ζ52+154+2ζ53+152+2ζ5+1-55    complex lifted from C23⋊F5
ρ2144-40000000000-11-1154+2ζ52+153+2ζ5+152+2ζ5+154+2ζ53+1-55    complex lifted from C23⋊F5
ρ228-800000000000-2-25225000000    orthogonal faithful
ρ238-800000000000-2252-25000000    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)

10000000
01000000
00100000
00010000
00001000
00000100
0000360400
00003929040
,
10000000
01000000
00100000
00010000
0000402500
000036100
0000529405
00003116161
,
400000000
040000000
004000000
000400000
00001000
000054000
00003611136
000090040
,
4040000000
3635000000
39396400000
00100000
00001000
00000100
00000010
00000001
,
03110000
21186400000
25252000000
010330000
0000400160
0000030395
00000110
00000173711

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

Export

Character table of (C2×D4)⋊F5 in TeX

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