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G = C42.F5order 320 = 26·5

1st non-split extension by C42 of F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.1F5, (C4×C20).1C4, C202Q8.1C2, (C2×Dic5).7D4, C10.3(C23⋊C4), C51(C42.3C4), (C2×Dic10).3C4, C2.6(D10.D4), Dic5.D4.1C2, (C2×Dic10).1C22, C22.10(C22⋊F5), (C2×C4).51(C2×F5), (C2×C20).97(C2×C4), (C2×C10).10(C22⋊C4), SmallGroup(320,193)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.F5
C1C5C10C2×C10C2×Dic5C2×Dic10Dic5.D4 — C42.F5
C5C10C2×C10C2×C20 — C42.F5
C1C2C22C2×C4C42

Generators and relations for C42.F5
 G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 282 in 60 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×4], Q8 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], C2×Q8 [×2], Dic5 [×3], C20 [×3], C2×C10, C4.10D4 [×2], C4⋊Q8, C5⋊C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C42.3C4, C4⋊Dic5 [×2], C4×C20, C22.F5 [×2], C2×Dic10 [×2], Dic5.D4 [×2], C202Q8, C42.F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C42.3C4, C22⋊F5, D10.D4, C42.F5

Character table of C42.F5

 class 12A2B4A4B4C4D4E4F58A8B8C8D10A10B10C20A20B20C20D20E20F20G20H20I20J20K20L
 size 112444202040440404040444444444444444
ρ111111111111111111111111111111    trivial
ρ2111-11-111-111-1-11111-1-11-1-1-1-1111-1-1    linear of order 2
ρ3111-11-111-11-111-1111-1-11-1-1-1-1111-1-1    linear of order 2
ρ41111111111-1-1-1-1111111111111111    linear of order 2
ρ5111-11-1-1-111-i-iii111-1-11-1-1-1-1111-1-1    linear of order 4
ρ6111111-1-1-11-ii-ii111111111111111    linear of order 4
ρ7111111-1-1-11i-ii-i111111111111111    linear of order 4
ρ8111-11-1-1-111ii-i-i111-1-11-1-1-1-1111-1-1    linear of order 4
ρ92220-20-2202000022200-20000-2-2-200    orthogonal lifted from D4
ρ102220-202-202000022200-20000-2-2-200    orthogonal lifted from D4
ρ1144-400000040000-4-44000000000000    orthogonal lifted from C23⋊C4
ρ12444-44-4000-10000-1-1-111-11111-1-1-111    orthogonal lifted from C2×F5
ρ13444444000-10000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ144440-40000-10000-1-1-1-5515-5-551115-5    orthogonal lifted from C22⋊F5
ρ154440-40000-10000-1-1-15-51-555-5111-55    orthogonal lifted from C22⋊F5
ρ1644-4000000-1000011-14ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ54+2ζ43ζ53435-5-543ζ52+2ζ43ζ5434ζ53+2ζ4ζ54    orthogonal lifted from D10.D4
ρ1744-4000000-1000011-143ζ54+2ζ43ζ53434ζ53+2ζ4ζ54-54ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ53+2ζ4ζ54-5554ζ54+2ζ4ζ52443ζ52+2ζ43ζ543    orthogonal lifted from D10.D4
ρ1844-4000000-1000011-143ζ52+2ζ43ζ5434ζ54+2ζ4ζ524-54ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ54+2ζ4ζ524-5554ζ53+2ζ4ζ5443ζ54+2ζ43ζ5343    orthogonal lifted from D10.D4
ρ1944-4000000-1000011-14ζ53+2ζ4ζ5443ζ52+2ζ43ζ543543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ52+2ζ43ζ5435-5-543ζ54+2ζ43ζ53434ζ54+2ζ4ζ524    orthogonal lifted from D10.D4
ρ204-4020-20004000000-4220-2-2-2-200022    symplectic lifted from C42.3C4, Schur index 2
ρ214-40-2020004000000-4-2-202222000-2-2    symplectic lifted from C42.3C4, Schur index 2
ρ224-4020-2000-10000-551ζ4ζ534ζ52+2ζ4ζ545352ζ43ζ54+2ζ43ζ5243ζ5435454ζ54+2ζ4ζ524ζ4ζ534ζ525352+1ζ43ζ5443ζ5545+143ζ5443ζ5545+14ζ534ζ525352+14ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543ζ43ζ54+2ζ43ζ5343ζ5435454ζ544ζ534ζ5245352    symplectic faithful, Schur index 2
ρ234-40-202000-10000-55143ζ5443ζ5545+14ζ534ζ525352+14ζ54+2ζ4ζ524ζ43ζ54+2ζ43ζ5343ζ5435454ζ544ζ534ζ5245352ζ4ζ534ζ52+2ζ4ζ545352ζ43ζ54+2ζ43ζ5243ζ5435454ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543ζ4ζ534ζ525352+1ζ43ζ5443ζ5545+1    symplectic faithful, Schur index 2
ρ244-4020-2000-10000-5514ζ544ζ534ζ5245352ζ43ζ54+2ζ43ζ5343ζ5435454ζ53+2ζ4ζ544ζ534ζ525352+143ζ5443ζ5545+1ζ43ζ5443ζ5545+1ζ4ζ534ζ525352+14ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ5343ζ43ζ54+2ζ43ζ5243ζ543545ζ4ζ534ζ52+2ζ4ζ545352    symplectic faithful, Schur index 2
ρ254-40-202000-10000-551ζ43ζ5443ζ5545+1ζ4ζ534ζ525352+14ζ53+2ζ4ζ54ζ43ζ54+2ζ43ζ5243ζ543545ζ4ζ534ζ52+2ζ4ζ5453524ζ544ζ534ζ5245352ζ43ζ54+2ζ43ζ5343ζ5435454ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ534ζ525352+143ζ5443ζ5545+1    symplectic faithful, Schur index 2
ρ264-40-202000-100005-51ζ4ζ534ζ525352+143ζ5443ζ5545+143ζ52+2ζ43ζ5434ζ544ζ534ζ5245352ζ43ζ54+2ζ43ζ5243ζ543545ζ43ζ54+2ζ43ζ5343ζ543545ζ4ζ534ζ52+2ζ4ζ54535243ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ54ζ43ζ5443ζ5545+14ζ534ζ525352+1    symplectic faithful, Schur index 2
ρ274-4020-2000-100005-51ζ43ζ54+2ζ43ζ5243ζ5435454ζ544ζ534ζ524535243ζ54+2ζ43ζ534343ζ5443ζ5545+1ζ4ζ534ζ525352+14ζ534ζ525352+1ζ43ζ5443ζ5545+143ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ524ζ4ζ534ζ52+2ζ4ζ545352ζ43ζ54+2ζ43ζ5343ζ543545    symplectic faithful, Schur index 2
ρ284-4020-2000-100005-51ζ43ζ54+2ζ43ζ5343ζ543545ζ4ζ534ζ52+2ζ4ζ54535243ζ52+2ζ43ζ543ζ43ζ5443ζ5545+14ζ534ζ525352+1ζ4ζ534ζ525352+143ζ5443ζ5545+143ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ544ζ544ζ534ζ5245352ζ43ζ54+2ζ43ζ5243ζ543545    symplectic faithful, Schur index 2
ρ294-40-202000-100005-514ζ534ζ525352+1ζ43ζ5443ζ5545+143ζ54+2ζ43ζ5343ζ4ζ534ζ52+2ζ4ζ545352ζ43ζ54+2ζ43ζ5343ζ543545ζ43ζ54+2ζ43ζ5243ζ5435454ζ544ζ534ζ524535243ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ5443ζ5545+1ζ4ζ534ζ525352+1    symplectic faithful, Schur index 2

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

Matrix representation of C42.F5 in GL4(𝔽41) generated by

1000
0100
003032
00911
,
303200
91100
00119
003230
,
344000
1000
0077
003440
,
0010
0001
222200
321900
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,30,9,0,0,32,11],[30,9,0,0,32,11,0,0,0,0,11,32,0,0,9,30],[34,1,0,0,40,0,0,0,0,0,7,34,0,0,7,40],[0,0,22,32,0,0,22,19,1,0,0,0,0,1,0,0] >;

C42.F5 in GAP, Magma, Sage, TeX

C_4^2.F_5
% in TeX

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

G:=SmallGroup(320,193);
// by ID

G=gap.SmallGroup(320,193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,184,1571,297,136,1684,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b^2,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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