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G = C42.2F5order 320 = 26·5

2nd non-split extension by C42 of F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.2F5, (C4×C20).2C4, (C2×D20).3C4, (C2×Dic5).8D4, C10.4(C23⋊C4), C51(C42.C4), C4.D20.2C2, Dic5.D41C2, C2.7(D10.D4), (C2×Dic10).2C22, C22.11(C22⋊F5), (C2×C4).52(C2×F5), (C2×C20).98(C2×C4), (C2×C10).11(C22⋊C4), SmallGroup(320,194)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.2F5
C1C5C10C2×C10C2×Dic5C2×Dic10Dic5.D4 — C42.2F5
C5C10C2×C10C2×C20 — C42.2F5
C1C2C22C2×C4C42

Generators and relations for C42.2F5
 G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 378 in 64 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×3], D4, Q8, C23, D5, C10, C10, C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C4.10D4 [×2], C4.4D4, C5⋊C8 [×2], Dic10, D20, C2×Dic5 [×2], C2×C20, C2×C20, C22×D5, C42.C4, D10⋊C4 [×2], C4×C20, C22.F5 [×2], C2×Dic10, C2×D20, Dic5.D4 [×2], C4.D20, C42.2F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C42.C4, C22⋊F5, D10.D4, C42.2F5

Character table of C42.2F5

 class 12A2B2C4A4B4C4D4E58A8B8C8D10A10B10C20A20B20C20D20E20F20G20H20I20J20K20L
 size 112404442020440404040444444444444444
ρ111111111111111111111111111111    trivial
ρ2111-1-1-11111-111-1111-1-11-1-1-1-1111-1-1    linear of order 2
ρ3111-1-1-111111-1-11111-1-11-1-1-1-1111-1-1    linear of order 2
ρ41111111111-1-1-1-1111111111111111    linear of order 2
ρ5111-1111-1-11-ii-ii111111111111111    linear of order 4
ρ61111-1-11-1-11ii-i-i111-1-11-1-1-1-1111-1-1    linear of order 4
ρ71111-1-11-1-11-i-iii111-1-11-1-1-1-1111-1-1    linear of order 4
ρ8111-1111-1-11i-ii-i111111111111111    linear of order 4
ρ9222000-2-222000022200-20000-2-2-200    orthogonal lifted from D4
ρ10222000-22-22000022200-20000-2-2-200    orthogonal lifted from D4
ρ114440-4-4400-10000-1-1-111-11111-1-1-111    orthogonal lifted from C2×F5
ρ1244-4000000400004-4-4000000000000    orthogonal lifted from C23⋊C4
ρ13444044400-10000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444-4000000-10000-1114ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ54+2ζ43ζ53435-5-543ζ52+2ζ43ζ5434ζ53+2ζ4ζ54    orthogonal lifted from D10.D4
ρ1544-4000000-10000-11143ζ52+2ζ43ζ5434ζ54+2ζ4ζ524-54ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ54+2ζ4ζ524-5554ζ53+2ζ4ζ5443ζ54+2ζ43ζ5343    orthogonal lifted from D10.D4
ρ16444000-400-10000-1-1-15-51-555-5111-55    orthogonal lifted from C22⋊F5
ρ17444000-400-10000-1-1-1-5515-5-551115-5    orthogonal lifted from C22⋊F5
ρ1844-4000000-10000-11143ζ54+2ζ43ζ53434ζ53+2ζ4ζ54-54ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ53+2ζ4ζ54-5554ζ54+2ζ4ζ52443ζ52+2ζ43ζ543    orthogonal lifted from D10.D4
ρ1944-4000000-10000-1114ζ53+2ζ4ζ5443ζ52+2ζ43ζ543543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ52+2ζ43ζ5435-5-543ζ54+2ζ43ζ53434ζ54+2ζ4ζ524    orthogonal lifted from D10.D4
ρ204-4002i-2i00040000-4002i2i0-2i-2i-2i-2i0002i2i    complex lifted from C42.C4
ρ214-4002i-2i000-100001-55ζ43ζ5443ζ543545ζ43ζ5343ζ524353524ζ53+2ζ4ζ54ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545ζ4ζ534ζ52453524ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ5343ζ43ζ5343ζ52435352ζ43ζ5443ζ543545    complex faithful
ρ224-400-2i2i000-1000015-5ζ4ζ534ζ5245352ζ4ζ544ζ5454543ζ52+2ζ43ζ543ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352ζ43ζ5443ζ54354543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ54ζ4ζ544ζ54545ζ4ζ534ζ5245352    complex faithful
ρ234-400-2i2i00040000-400-2i-2i02i2i2i2i000-2i-2i    complex lifted from C42.C4
ρ244-400-2i2i000-100001-55ζ4ζ544ζ54545ζ4ζ534ζ52453524ζ54+2ζ4ζ524ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545ζ43ζ5343ζ524353524ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543ζ4ζ534ζ5245352ζ4ζ544ζ54545    complex faithful
ρ254-4002i-2i000-1000015-5ζ43ζ5343ζ52435352ζ43ζ5443ζ54354543ζ54+2ζ43ζ5343ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352ζ4ζ544ζ5454543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ524ζ43ζ5443ζ543545ζ43ζ5343ζ52435352    complex faithful
ρ264-400-2i2i000-1000015-5ζ4ζ534ζ5245352ζ4ζ544ζ5454543ζ54+2ζ43ζ5343ζ43ζ5443ζ543545ζ43ζ5343ζ52435352ζ43ζ5343ζ52435352ζ43ζ5443ζ54354543ζ52+2ζ43ζ5434ζ53+2ζ4ζ544ζ54+2ζ4ζ524ζ4ζ544ζ54545ζ4ζ534ζ5245352    complex faithful
ρ274-400-2i2i000-100001-55ζ4ζ544ζ54545ζ4ζ534ζ52453524ζ53+2ζ4ζ54ζ43ζ5343ζ52435352ζ43ζ5443ζ543545ζ43ζ5443ζ543545ζ43ζ5343ζ524353524ζ54+2ζ4ζ52443ζ52+2ζ43ζ54343ζ54+2ζ43ζ5343ζ4ζ534ζ5245352ζ4ζ544ζ54545    complex faithful
ρ284-4002i-2i000-100001-55ζ43ζ5443ζ543545ζ43ζ5343ζ524353524ζ54+2ζ4ζ524ζ4ζ534ζ5245352ζ4ζ544ζ54545ζ4ζ544ζ54545ζ4ζ534ζ52453524ζ53+2ζ4ζ5443ζ54+2ζ43ζ534343ζ52+2ζ43ζ543ζ43ζ5343ζ52435352ζ43ζ5443ζ543545    complex faithful
ρ294-4002i-2i000-1000015-5ζ43ζ5343ζ52435352ζ43ζ5443ζ54354543ζ52+2ζ43ζ543ζ4ζ544ζ54545ζ4ζ534ζ5245352ζ4ζ534ζ5245352ζ4ζ544ζ5454543ζ54+2ζ43ζ53434ζ54+2ζ4ζ5244ζ53+2ζ4ζ54ζ43ζ5443ζ543545ζ43ζ5343ζ52435352    complex faithful

Smallest permutation representation of C42.2F5
On 80 points
Generators in S80
(1 7)(2 57 6 61)(3 5)(4 59 8 63)(9 68 13 72)(10 79)(11 70 15 66)(12 77)(14 75)(16 73)(17 51 21 55)(18 47)(19 53 23 49)(20 45)(22 43)(24 41)(25 71)(26 74 30 78)(27 69)(28 76 32 80)(29 67)(31 65)(33 42 37 46)(34 52)(35 44 39 48)(36 50)(38 56)(40 54)(58 60)(62 64)
(1 62 5 58)(2 59 6 63)(3 60 7 64)(4 57 8 61)(9 26 13 30)(10 27 14 31)(11 32 15 28)(12 25 16 29)(17 33 21 37)(18 34 22 38)(19 39 23 35)(20 40 24 36)(41 50 45 54)(42 55 46 51)(43 56 47 52)(44 53 48 49)(65 79 69 75)(66 76 70 80)(67 77 71 73)(68 74 72 78)
(1 12 18 41 79)(2 42 13 80 19)(3 73 43 20 14)(4 21 74 15 44)(5 16 22 45 75)(6 46 9 76 23)(7 77 47 24 10)(8 17 78 11 48)(25 34 50 69 62)(26 70 35 63 51)(27 64 71 52 36)(28 53 57 37 72)(29 38 54 65 58)(30 66 39 59 55)(31 60 67 56 40)(32 49 61 33 68)
(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)| (1,7)(2,57,6,61)(3,5)(4,59,8,63)(9,68,13,72)(10,79)(11,70,15,66)(12,77)(14,75)(16,73)(17,51,21,55)(18,47)(19,53,23,49)(20,45)(22,43)(24,41)(25,71)(26,74,30,78)(27,69)(28,76,32,80)(29,67)(31,65)(33,42,37,46)(34,52)(35,44,39,48)(36,50)(38,56)(40,54)(58,60)(62,64), (1,62,5,58)(2,59,6,63)(3,60,7,64)(4,57,8,61)(9,26,13,30)(10,27,14,31)(11,32,15,28)(12,25,16,29)(17,33,21,37)(18,34,22,38)(19,39,23,35)(20,40,24,36)(41,50,45,54)(42,55,46,51)(43,56,47,52)(44,53,48,49)(65,79,69,75)(66,76,70,80)(67,77,71,73)(68,74,72,78), (1,12,18,41,79)(2,42,13,80,19)(3,73,43,20,14)(4,21,74,15,44)(5,16,22,45,75)(6,46,9,76,23)(7,77,47,24,10)(8,17,78,11,48)(25,34,50,69,62)(26,70,35,63,51)(27,64,71,52,36)(28,53,57,37,72)(29,38,54,65,58)(30,66,39,59,55)(31,60,67,56,40)(32,49,61,33,68), (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( (1,7)(2,57,6,61)(3,5)(4,59,8,63)(9,68,13,72)(10,79)(11,70,15,66)(12,77)(14,75)(16,73)(17,51,21,55)(18,47)(19,53,23,49)(20,45)(22,43)(24,41)(25,71)(26,74,30,78)(27,69)(28,76,32,80)(29,67)(31,65)(33,42,37,46)(34,52)(35,44,39,48)(36,50)(38,56)(40,54)(58,60)(62,64), (1,62,5,58)(2,59,6,63)(3,60,7,64)(4,57,8,61)(9,26,13,30)(10,27,14,31)(11,32,15,28)(12,25,16,29)(17,33,21,37)(18,34,22,38)(19,39,23,35)(20,40,24,36)(41,50,45,54)(42,55,46,51)(43,56,47,52)(44,53,48,49)(65,79,69,75)(66,76,70,80)(67,77,71,73)(68,74,72,78), (1,12,18,41,79)(2,42,13,80,19)(3,73,43,20,14)(4,21,74,15,44)(5,16,22,45,75)(6,46,9,76,23)(7,77,47,24,10)(8,17,78,11,48)(25,34,50,69,62)(26,70,35,63,51)(27,64,71,52,36)(28,53,57,37,72)(29,38,54,65,58)(30,66,39,59,55)(31,60,67,56,40)(32,49,61,33,68), (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([(1,7),(2,57,6,61),(3,5),(4,59,8,63),(9,68,13,72),(10,79),(11,70,15,66),(12,77),(14,75),(16,73),(17,51,21,55),(18,47),(19,53,23,49),(20,45),(22,43),(24,41),(25,71),(26,74,30,78),(27,69),(28,76,32,80),(29,67),(31,65),(33,42,37,46),(34,52),(35,44,39,48),(36,50),(38,56),(40,54),(58,60),(62,64)], [(1,62,5,58),(2,59,6,63),(3,60,7,64),(4,57,8,61),(9,26,13,30),(10,27,14,31),(11,32,15,28),(12,25,16,29),(17,33,21,37),(18,34,22,38),(19,39,23,35),(20,40,24,36),(41,50,45,54),(42,55,46,51),(43,56,47,52),(44,53,48,49),(65,79,69,75),(66,76,70,80),(67,77,71,73),(68,74,72,78)], [(1,12,18,41,79),(2,42,13,80,19),(3,73,43,20,14),(4,21,74,15,44),(5,16,22,45,75),(6,46,9,76,23),(7,77,47,24,10),(8,17,78,11,48),(25,34,50,69,62),(26,70,35,63,51),(27,64,71,52,36),(28,53,57,37,72),(29,38,54,65,58),(30,66,39,59,55),(31,60,67,56,40),(32,49,61,33,68)], [(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 C42.2F5 in GL4(𝔽41) generated by

174000
12400
0090
0009
,
303200
91100
003032
00911
,
344000
1000
0077
003440
,
0010
0001
174000
32400
G:=sub<GL(4,GF(41))| [17,1,0,0,40,24,0,0,0,0,9,0,0,0,0,9],[30,9,0,0,32,11,0,0,0,0,30,9,0,0,32,11],[34,1,0,0,40,0,0,0,0,0,7,34,0,0,7,40],[0,0,17,3,0,0,40,24,1,0,0,0,0,1,0,0] >;

C42.2F5 in GAP, Magma, Sage, TeX

C_4^2._2F_5
% in TeX

G:=Group("C4^2.2F5");
// GroupNames label

G:=SmallGroup(320,194);
// by ID

G=gap.SmallGroup(320,194);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,555,184,675,297,136,1684,6278,3156]);
// 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,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of C42.2F5 in TeX

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