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G = C422F5order 320 = 26·5

2nd semidirect product of C42 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C422F5, (C4×C20)⋊2C4, C51(C423C4), (C2×Dic10)⋊3C4, (C22×D5).8D4, C10.2(C23⋊C4), C4.D20.1C2, (C2×D20).2C22, D10.D4.1C2, C22.9(C22⋊F5), C2.5(D10.D4), (C2×C4).50(C2×F5), (C2×C20).96(C2×C4), (C2×C10).9(C22⋊C4), SmallGroup(320,192)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C422F5
C1C5C10C2×C10C22×D5C2×D20D10.D4 — C422F5
C5C10C2×C10C2×C20 — C422F5
C1C2C22C2×C4C42

Generators and relations for C422F5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 474 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], D5 [×2], C10, C10, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5, C20 [×2], F5 [×2], D10 [×4], C2×C10, C23⋊C4 [×2], C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×F5 [×2], C22×D5 [×2], C423C4, D10⋊C4 [×2], C4×C20, C22⋊F5 [×2], C2×Dic10, C2×D20, D10.D4 [×2], C4.D20, C422F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C423C4, C22⋊F5, D10.D4, C422F5

Character table of C422F5

 class 12A2B2C2D4A4B4C4D4E4F4G4H510A10B10C20A20B20C20D20E20F20G20H20I20J20K20L
 size 112202044440404040404444444444444444
ρ111111111111111111111111111111    trivial
ρ211111-11-1-1-111-11111-111-1-1-1-111-1-1-1    linear of order 2
ρ311111111-11-1-1-11111111111111111    linear of order 2
ρ411111-11-11-1-1-111111-111-1-1-1-111-1-1-1    linear of order 2
ρ5111-1-1111i-1-ii-i1111111111111111    linear of order 4
ρ6111-1-1-11-1-i1-iii1111-111-1-1-1-111-1-1-1    linear of order 4
ρ7111-1-1111-i-1i-ii1111111111111111    linear of order 4
ρ8111-1-1-11-1i1i-i-i1111-111-1-1-1-111-1-1-1    linear of order 4
ρ92222-20-200000022220-2-20000-2-2000    orthogonal lifted from D4
ρ10222-220-200000022220-2-20000-2-2000    orthogonal lifted from D4
ρ1144400-44-400000-1-1-1-11-1-11111-1-1111    orthogonal lifted from C2×F5
ρ1244-400000000004-4-44000000000000    orthogonal lifted from C23⋊C4
ρ134440044400000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444-40000000000-111-14ζ54+2ζ4ζ524-554ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ54+2ζ4ζ5245-54ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543    orthogonal lifted from D10.D4
ρ15444000-4000000-1-1-1-1-511-555-511-555    orthogonal lifted from C22⋊F5
ρ16444000-4000000-1-1-1-15115-5-55115-5-5    orthogonal lifted from C22⋊F5
ρ1744-40000000000-111-14ζ53+2ζ4ζ54-554ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ53+2ζ4ζ545-54ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ5343    orthogonal lifted from D10.D4
ρ1844-40000000000-111-143ζ52+2ζ43ζ5435-543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ52+2ζ43ζ543-5543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ54    orthogonal lifted from D10.D4
ρ1944-40000000000-111-143ζ54+2ζ43ζ53435-543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343-5543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ524    orthogonal lifted from D10.D4
ρ204-40002i0-2i00000400-42i00-2i-2i-2i-2i002i2i2i    complex lifted from C423C4
ρ214-40002i0-2i00000-1-551ζ43ζ5343ζ524353524ζ53+2ζ4ζ5443ζ54+2ζ43ζ5343ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545ζ4ζ534ζ524535243ζ52+2ζ43ζ5434ζ54+2ζ4ζ524ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545    complex faithful
ρ224-40002i0-2i00000-1-551ζ43ζ5343ζ524353524ζ54+2ζ4ζ52443ζ52+2ζ43ζ543ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545ζ4ζ534ζ524535243ζ54+2ζ43ζ53434ζ53+2ζ4ζ54ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545    complex faithful
ρ234-4000-2i02i00000-15-51ζ4ζ544ζ5454543ζ52+2ζ43ζ5434ζ53+2ζ4ζ54ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352ζ43ζ5443ζ5435454ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352    complex faithful
ρ244-4000-2i02i00000-15-51ζ4ζ544ζ5454543ζ54+2ζ43ζ53434ζ54+2ζ4ζ524ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352ζ43ζ5443ζ5435454ζ53+2ζ4ζ5443ζ52+2ζ43ζ543ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352    complex faithful
ρ254-4000-2i02i00000-1-551ζ4ζ534ζ52453524ζ54+2ζ4ζ52443ζ52+2ζ43ζ543ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545ζ43ζ5343ζ5243535243ζ54+2ζ43ζ53434ζ53+2ζ4ζ54ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545    complex faithful
ρ264-40002i0-2i00000-15-51ζ43ζ5443ζ54354543ζ54+2ζ43ζ53434ζ54+2ζ4ζ524ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352ζ4ζ544ζ545454ζ53+2ζ4ζ5443ζ52+2ζ43ζ543ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352    complex faithful
ρ274-4000-2i02i00000400-4-2i002i2i2i2i00-2i-2i-2i    complex lifted from C423C4
ρ284-4000-2i02i00000-1-551ζ4ζ534ζ52453524ζ53+2ζ4ζ5443ζ54+2ζ43ζ5343ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545ζ43ζ5343ζ5243535243ζ52+2ζ43ζ5434ζ54+2ζ4ζ524ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545    complex faithful
ρ294-40002i0-2i00000-15-51ζ43ζ5443ζ54354543ζ52+2ζ43ζ5434ζ53+2ζ4ζ54ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352ζ4ζ544ζ545454ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352    complex faithful

Smallest permutation representation of C422F5
On 80 points
Generators in S80
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31 26 36)(22 33 30 39)(23 35 29 37)(24 32 28 40)(25 34 27 38)(41 61 56 71)(42 63 60 74)(43 65 59 72)(44 62 58 75)(45 64 57 73)(46 66 51 76)(47 68 55 79)(48 70 54 77)(49 67 53 80)(50 69 52 78)

G:=sub<Sym(80)| (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,56,71)(42,63,60,74)(43,65,59,72)(44,62,58,75)(45,64,57,73)(46,66,51,76)(47,68,55,79)(48,70,54,77)(49,67,53,80)(50,69,52,78)>;

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

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

Matrix representation of C422F5 in GL4(𝔽41) generated by

269729
12382119
2234192
39203217
,
36740
1478
33343740
1343538
,
40404040
1000
0100
0010
,
1000
0001
0100
40404040
G:=sub<GL(4,GF(41))| [26,12,22,39,9,38,34,20,7,21,19,32,29,19,2,17],[3,1,33,1,6,4,34,34,7,7,37,35,40,8,40,38],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[1,0,0,40,0,0,1,40,0,0,0,40,0,1,0,40] >;

C422F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2F_5
% in TeX

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

G:=SmallGroup(320,192);
// by ID

G=gap.SmallGroup(320,192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,555,675,297,136,1684,6278,3156]);
// 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,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of C422F5 in TeX

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