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G = C42⋊F5order 320 = 26·5

1st semidirect product of C42 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C421F5, (C4×C20)⋊1C4, (C2×D20)⋊3C4, C51(C42⋊C4), C204D4.1C2, (C22×D5).7D4, D10.D41C2, C10.1(C23⋊C4), (C2×D20).1C22, C22.8(C22⋊F5), C2.4(D10.D4), (C2×C4).49(C2×F5), (C2×C20).95(C2×C4), (C2×C10).8(C22⋊C4), SmallGroup(320,191)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42⋊F5
C1C5C10C2×C10C22×D5C2×D20D10.D4 — C42⋊F5
C5C10C2×C10C2×C20 — C42⋊F5
C1C2C22C2×C4C42

Generators and relations for C42⋊F5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 666 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 [×3], C10, C10, C42, C22⋊C4 [×2], C2×D4 [×4], C20 [×3], F5 [×2], D10 [×7], C2×C10, C23⋊C4 [×2], C41D4, D20 [×6], C2×C20, C2×C20, C2×F5 [×2], C22×D5 [×2], C22×D5, C42⋊C4, C4×C20, C22⋊F5 [×2], C2×D20 [×2], C2×D20 [×2], D10.D4 [×2], C204D4, C42⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C42⋊C4, C22⋊F5, D10.D4, C42⋊F5

Character table of C42⋊F5

 class 12A2B2C2D2E4A4B4C4D4E4F4G510A10B10C20A20B20C20D20E20F20G20H20I20J20K20L
 size 112202040444404040404444444444444444
ρ111111111111111111111111111111    trivial
ρ211111-1-1-11-111-11111-1-11-1-1-1-1111-1-1    linear of order 2
ρ311111-1-1-111-1-111111-1-11-1-1-1-1111-1-1    linear of order 2
ρ4111111111-1-1-1-11111111111111111    linear of order 2
ρ5111-1-1-1111-ii-ii1111111111111111    linear of order 4
ρ6111-1-11-1-11ii-i-i1111-1-11-1-1-1-1111-1-1    linear of order 4
ρ7111-1-11-1-11-i-iii1111-1-11-1-1-1-1111-1-1    linear of order 4
ρ8111-1-1-1111i-ii-i1111111111111111    linear of order 4
ρ92222-2000-20000222200-20000-2-2-200    orthogonal lifted from D4
ρ10222-22000-20000222200-20000-2-2-200    orthogonal lifted from D4
ρ1144-4000000000044-4-4000000000000    orthogonal lifted from C23⋊C4
ρ124-40000-22000004-400-2-202222000-2-2    orthogonal lifted from C42⋊C4
ρ13444000-4-440000-1-1-1-111-11111-1-1-111    orthogonal lifted from C2×F5
ρ144-400002-2000004-400220-2-2-2-200022    orthogonal lifted from C42⋊C4
ρ154440004440000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1644400000-40000-1-1-1-15-51-555-5111-55    orthogonal lifted from C22⋊F5
ρ1744400000-40000-1-1-1-1-5515-5-551115-5    orthogonal lifted from C22⋊F5
ρ184-40000-2200000-115-5ζ43ζ5443ζ5545+1ζ4ζ534ζ525352+14ζ54+2ζ4ζ534ζ4ζ534ζ525352-1ζ43ζ5443ζ5545-143ζ5443ζ5545-14ζ534ζ525352-143ζ53+2ζ43ζ54343ζ54+2ζ43ζ52434ζ52+2ζ4ζ544ζ534ζ525352+143ζ5443ζ5545+1    orthogonal faithful
ρ1944-40000000000-1-11143ζ53+2ζ43ζ5434ζ52+2ζ4ζ54-54ζ54+2ζ4ζ53443ζ54+2ζ43ζ524343ζ53+2ζ43ζ5434ζ52+2ζ4ζ5455-54ζ54+2ζ4ζ53443ζ54+2ζ43ζ5243    orthogonal lifted from D10.D4
ρ204-40000-2200000-115-543ζ5443ζ5545+14ζ534ζ525352+14ζ52+2ζ4ζ544ζ534ζ525352-143ζ5443ζ5545-1ζ43ζ5443ζ5545-1ζ4ζ534ζ525352-143ζ54+2ζ43ζ524343ζ53+2ζ43ζ5434ζ54+2ζ4ζ534ζ4ζ534ζ525352+1ζ43ζ5443ζ5545+1    orthogonal faithful
ρ2144-40000000000-1-11143ζ54+2ζ43ζ52434ζ54+2ζ4ζ534-54ζ52+2ζ4ζ5443ζ53+2ζ43ζ54343ζ54+2ζ43ζ52434ζ54+2ζ4ζ53455-54ζ52+2ζ4ζ5443ζ53+2ζ43ζ543    orthogonal lifted from D10.D4
ρ2244-40000000000-1-1114ζ54+2ζ4ζ53443ζ53+2ζ43ζ543543ζ54+2ζ43ζ52434ζ52+2ζ4ζ544ζ54+2ζ4ζ53443ζ53+2ζ43ζ543-5-5543ζ54+2ζ43ζ52434ζ52+2ζ4ζ54    orthogonal lifted from D10.D4
ρ234-40000-2200000-11-554ζ534ζ525352+1ζ43ζ5443ζ5545+143ζ54+2ζ43ζ5243ζ43ζ5443ζ5545-14ζ534ζ525352-1ζ4ζ534ζ525352-143ζ5443ζ5545-14ζ54+2ζ4ζ5344ζ52+2ζ4ζ5443ζ53+2ζ43ζ54343ζ5443ζ5545+1ζ4ζ534ζ525352+1    orthogonal faithful
ρ244-400002-200000-11-554ζ534ζ525352-1ζ43ζ5443ζ5545-143ζ53+2ζ43ζ543ζ43ζ5443ζ5545+14ζ534ζ525352+1ζ4ζ534ζ525352+143ζ5443ζ5545+14ζ52+2ζ4ζ544ζ54+2ζ4ζ53443ζ54+2ζ43ζ524343ζ5443ζ5545-1ζ4ζ534ζ525352-1    orthogonal faithful
ρ254-400002-200000-115-5ζ43ζ5443ζ5545-1ζ4ζ534ζ525352-14ζ52+2ζ4ζ54ζ4ζ534ζ525352+1ζ43ζ5443ζ5545+143ζ5443ζ5545+14ζ534ζ525352+143ζ54+2ζ43ζ524343ζ53+2ζ43ζ5434ζ54+2ζ4ζ5344ζ534ζ525352-143ζ5443ζ5545-1    orthogonal faithful
ρ264-40000-2200000-11-55ζ4ζ534ζ525352+143ζ5443ζ5545+143ζ53+2ζ43ζ54343ζ5443ζ5545-1ζ4ζ534ζ525352-14ζ534ζ525352-1ζ43ζ5443ζ5545-14ζ52+2ζ4ζ544ζ54+2ζ4ζ53443ζ54+2ζ43ζ5243ζ43ζ5443ζ5545+14ζ534ζ525352+1    orthogonal faithful
ρ2744-40000000000-1-1114ζ52+2ζ4ζ5443ζ54+2ζ43ζ5243543ζ53+2ζ43ζ5434ζ54+2ζ4ζ5344ζ52+2ζ4ζ5443ζ54+2ζ43ζ5243-5-5543ζ53+2ζ43ζ5434ζ54+2ζ4ζ534    orthogonal lifted from D10.D4
ρ284-400002-200000-115-543ζ5443ζ5545-14ζ534ζ525352-14ζ54+2ζ4ζ5344ζ534ζ525352+143ζ5443ζ5545+1ζ43ζ5443ζ5545+1ζ4ζ534ζ525352+143ζ53+2ζ43ζ54343ζ54+2ζ43ζ52434ζ52+2ζ4ζ54ζ4ζ534ζ525352-1ζ43ζ5443ζ5545-1    orthogonal faithful
ρ294-400002-200000-11-55ζ4ζ534ζ525352-143ζ5443ζ5545-143ζ54+2ζ43ζ524343ζ5443ζ5545+1ζ4ζ534ζ525352+14ζ534ζ525352+1ζ43ζ5443ζ5545+14ζ54+2ζ4ζ5344ζ52+2ζ4ζ5443ζ53+2ζ43ζ543ζ43ζ5443ζ5545-14ζ534ζ525352-1    orthogonal faithful

Smallest permutation representation of C42⋊F5
On 40 points
Generators in S40
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 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)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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([(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(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 C42⋊F5 in GL4(𝔽41) generated by

40000
04000
20243913
2630282
,
302800
221100
3723913
2123282
,
404000
8700
3704035
3636635
,
1192020
71813
27132818
082025
G:=sub<GL(4,GF(41))| [40,0,20,26,0,40,24,30,0,0,39,28,0,0,13,2],[30,22,37,21,28,11,2,23,0,0,39,28,0,0,13,2],[40,8,37,36,40,7,0,36,0,0,40,6,0,0,35,35],[11,7,27,0,9,18,13,8,20,1,28,20,20,3,18,25] >;

C42⋊F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes F_5
% in TeX

G:=Group("C4^2:F5");
// GroupNames label

G:=SmallGroup(320,191);
// by ID

G=gap.SmallGroup(320,191);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,1571,297,136,1684,6278,3156]);
// Polycyclic

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

Export

Character table of C42⋊F5 in TeX

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