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## G = C42.2F5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.2F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — Dic5.D4 — C42.2F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42.2F5
 Upper central C1 — C2 — C22 — C2×C4 — C42

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, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C4.10D4, C4.4D4, C5⋊C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.C4, D10⋊C4, C4×C20, C22.F5, C2×Dic10, C2×D20, Dic5.D4, C4.D20, C42.2F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42.C4, C22⋊F5, D10.D4, C42.2F5

Character table of C42.2F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 10A 10B 10C 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L size 1 1 2 40 4 4 4 20 20 4 40 40 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 -1 1 -i i -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 -1 1 i i -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 -1 1 -i -i 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 -1 1 i -i 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 0 0 0 -2 -2 2 2 0 0 0 0 2 2 2 0 0 -2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 0 0 -2 2 -2 2 0 0 0 0 2 2 2 0 0 -2 0 0 0 0 -2 -2 -2 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 -4 -4 4 0 0 -1 0 0 0 0 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ12 4 4 -4 0 0 0 0 0 0 4 0 0 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ13 4 4 4 0 4 4 4 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 -1 1 1 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 √5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 √5 -√5 -√5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 orthogonal lifted from D10.D4 ρ15 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 -1 1 1 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 -√5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 -√5 √5 √5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 orthogonal lifted from D10.D4 ρ16 4 4 4 0 0 0 -4 0 0 -1 0 0 0 0 -1 -1 -1 √5 -√5 1 -√5 √5 √5 -√5 1 1 1 -√5 √5 orthogonal lifted from C22⋊F5 ρ17 4 4 4 0 0 0 -4 0 0 -1 0 0 0 0 -1 -1 -1 -√5 √5 1 √5 -√5 -√5 √5 1 1 1 √5 -√5 orthogonal lifted from C22⋊F5 ρ18 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 -1 1 1 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 -√5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 -√5 √5 √5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 orthogonal lifted from D10.D4 ρ19 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 -1 1 1 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 √5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 √5 -√5 -√5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 orthogonal lifted from D10.D4 ρ20 4 -4 0 0 2i -2i 0 0 0 4 0 0 0 0 -4 0 0 2i 2i 0 -2i -2i -2i -2i 0 0 0 2i 2i complex lifted from C42.C4 ρ21 4 -4 0 0 2i -2i 0 0 0 -1 0 0 0 0 1 -√5 √5 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 complex faithful ρ22 4 -4 0 0 -2i 2i 0 0 0 -1 0 0 0 0 1 √5 -√5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 complex faithful ρ23 4 -4 0 0 -2i 2i 0 0 0 4 0 0 0 0 -4 0 0 -2i -2i 0 2i 2i 2i 2i 0 0 0 -2i -2i complex lifted from C42.C4 ρ24 4 -4 0 0 -2i 2i 0 0 0 -1 0 0 0 0 1 -√5 √5 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 complex faithful ρ25 4 -4 0 0 2i -2i 0 0 0 -1 0 0 0 0 1 √5 -√5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 complex faithful ρ26 4 -4 0 0 -2i 2i 0 0 0 -1 0 0 0 0 1 √5 -√5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 complex faithful ρ27 4 -4 0 0 -2i 2i 0 0 0 -1 0 0 0 0 1 -√5 √5 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 complex faithful ρ28 4 -4 0 0 2i -2i 0 0 0 -1 0 0 0 0 1 -√5 √5 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 complex faithful ρ29 4 -4 0 0 2i -2i 0 0 0 -1 0 0 0 0 1 √5 -√5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 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

 17 40 0 0 1 24 0 0 0 0 9 0 0 0 0 9
,
 30 32 0 0 9 11 0 0 0 0 30 32 0 0 9 11
,
 34 40 0 0 1 0 0 0 0 0 7 7 0 0 34 40
,
 0 0 1 0 0 0 0 1 17 40 0 0 3 24 0 0
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

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