metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C22×C4)⋊1F5, (C22×C20)⋊5C4, C23.D5⋊6C4, C23⋊F5.2C2, C2.4(C23⋊F5), C23.18(C2×F5), C23.F5.2C2, (C2×Dic5).11D4, (C22×D5).11D4, C5⋊2(C23.D4), C10.13(C23⋊C4), C22.18(C22⋊F5), C23.23D10.1C2, (C22×C10).45(C2×C4), (C2×C5⋊D4).85C22, (C2×C10).29(C22⋊C4), SmallGroup(320,254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C22×C4)⋊F5
G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=abc2, bc=cb, bd=db, ebe-1=bc2, cd=dc, ece-1=abc-1, ede-1=d3 >
Subgroups: 378 in 68 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic5, C20, F5, D10, C2×C10, C2×C10, C23⋊C4, C4.D4, C22.D4, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×F5, C22×D5, C22×C10, C23.D4, C10.D4, D10⋊C4, C23.D5, C22.F5, C22⋊F5, C2×C5⋊D4, C22×C20, C23⋊F5, C23.F5, C23.23D10, (C22×C4)⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C23.D4, C22⋊F5, C23⋊F5, (C22×C4)⋊F5
Character table of (C22×C4)⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 2 | 4 | 20 | 4 | 4 | 20 | 40 | 40 | 40 | 4 | 40 | 40 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | 1 | i | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | 1 | -i | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | i | -1 | -i | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -i | -1 | i | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ12 | 4 | 4 | 4 | 4 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ14 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -√5 | -√5 | √5 | -√5 | -√5 | √5 | √5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | √5 | √5 | -√5 | √5 | √5 | -√5 | -√5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | -√5 | √5 | √5 | -√5 | 1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ52+2ζ5+1 | complex lifted from C23⋊F5 |
| ρ17 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | -√5 | √5 | √5 | -√5 | 1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ53+1 | complex lifted from C23⋊F5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 2i | -2i | -2i | -2i | -2i | 2i | complex lifted from C23.D4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | -2i | 2i | 2i | 2i | 2i | -2i | complex lifted from C23.D4 |
| ρ20 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | √5 | -√5 | -√5 | √5 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ53+2ζ5+1 | complex lifted from C23⋊F5 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | √5 | -√5 | -√5 | √5 | 1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ52+1 | complex lifted from C23⋊F5 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -√5 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | √5 | 2ζ43ζ5+ζ43 | 2ζ43ζ54+ζ43 | 2ζ43ζ52+ζ43 | 2ζ4ζ5+ζ4 | 2ζ4ζ54+ζ4 | 2ζ4ζ52+ζ4 | 2ζ4ζ53+ζ4 | 2ζ43ζ53+ζ43 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | √5 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | -√5 | 2ζ4ζ52+ζ4 | 2ζ4ζ53+ζ4 | 2ζ4ζ54+ζ4 | 2ζ43ζ52+ζ43 | 2ζ43ζ53+ζ43 | 2ζ43ζ54+ζ43 | 2ζ43ζ5+ζ43 | 2ζ4ζ5+ζ4 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -√5 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | √5 | 2ζ4ζ54+ζ4 | 2ζ4ζ5+ζ4 | 2ζ4ζ53+ζ4 | 2ζ43ζ54+ζ43 | 2ζ43ζ5+ζ43 | 2ζ43ζ53+ζ43 | 2ζ43ζ52+ζ43 | 2ζ4ζ52+ζ4 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | √5 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | -√5 | 2ζ43ζ53+ζ43 | 2ζ43ζ52+ζ43 | 2ζ43ζ5+ζ43 | 2ζ4ζ53+ζ4 | 2ζ4ζ52+ζ4 | 2ζ4ζ5+ζ4 | 2ζ4ζ54+ζ4 | 2ζ43ζ54+ζ43 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | √5 | 2ζ54+2ζ53+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | -√5 | 2ζ4ζ53+ζ4 | 2ζ4ζ52+ζ4 | 2ζ4ζ5+ζ4 | 2ζ43ζ53+ζ43 | 2ζ43ζ52+ζ43 | 2ζ43ζ5+ζ43 | 2ζ43ζ54+ζ43 | 2ζ4ζ54+ζ4 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | √5 | 2ζ52+2ζ5+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | -√5 | 2ζ43ζ52+ζ43 | 2ζ43ζ53+ζ43 | 2ζ43ζ54+ζ43 | 2ζ4ζ52+ζ4 | 2ζ4ζ53+ζ4 | 2ζ4ζ54+ζ4 | 2ζ4ζ5+ζ4 | 2ζ43ζ5+ζ43 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -√5 | 2ζ54+2ζ52+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | √5 | 2ζ43ζ54+ζ43 | 2ζ43ζ5+ζ43 | 2ζ43ζ53+ζ43 | 2ζ4ζ54+ζ4 | 2ζ4ζ5+ζ4 | 2ζ4ζ53+ζ4 | 2ζ4ζ52+ζ4 | 2ζ43ζ52+ζ43 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -√5 | 2ζ53+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | √5 | 2ζ4ζ5+ζ4 | 2ζ4ζ54+ζ4 | 2ζ4ζ52+ζ4 | 2ζ43ζ5+ζ43 | 2ζ43ζ54+ζ43 | 2ζ43ζ52+ζ43 | 2ζ43ζ53+ζ43 | 2ζ4ζ53+ζ4 | complex faithful |
►On 80 points
Generators in S80
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 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 51 76)(42 63 55 79)(43 65 54 77)(44 62 53 80)(45 64 52 78)(46 66 56 71)(47 68 60 74)(48 70 59 72)(49 67 58 75)(50 69 57 73)
G:=sub<Sym(80)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,46,6,41)(2,47,7,42)(3,48,8,43)(4,49,9,44)(5,50,10,45)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (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)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,46,6,41)(2,47,7,42)(3,48,8,43)(4,49,9,44)(5,50,10,45)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (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)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,46,6,41),(2,47,7,42),(3,48,8,43),(4,49,9,44),(5,50,10,45),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)], [(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),(41,61,51,76),(42,63,55,79),(43,65,54,77),(44,62,53,80),(45,64,52,78),(46,66,56,71),(47,68,60,74),(48,70,59,72),(49,67,58,75),(50,69,57,73)]])
Matrix representation of (C22×C4)⋊F5 ►in GL4(𝔽41) generated by
| 5 | 10 | 32 | 19 |
| 22 | 27 | 32 | 13 |
| 28 | 9 | 14 | 19 |
| 22 | 9 | 31 | 36 |
| 22 | 0 | 3 | 3 |
| 38 | 19 | 38 | 0 |
| 0 | 38 | 19 | 38 |
| 3 | 3 | 0 | 22 |
| 22 | 1 | 31 | 38 |
| 3 | 25 | 4 | 34 |
| 7 | 10 | 32 | 11 |
| 30 | 37 | 40 | 21 |
| 40 | 40 | 40 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 40 | 40 | 40 |
G:=sub<GL(4,GF(41))| [5,22,28,22,10,27,9,9,32,32,14,31,19,13,19,36],[22,38,0,3,0,19,38,3,3,38,19,0,3,0,38,22],[22,3,7,30,1,25,10,37,31,4,32,40,38,34,11,21],[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] >;
(C22×C4)⋊F5 in GAP, Magma, Sage, TeX
(C_2^2\times C_4)\rtimes F_5
% in TeX
G:=Group("(C2^2xC4):F5"); // GroupNames label
G:=SmallGroup(320,254);
// by ID
G=gap.SmallGroup(320,254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,675,297,1684,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c^2,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^2,c*d=d*c,e*c*e^-1=a*b*c^-1,e*d*e^-1=d^3>;
// generators/relations
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