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## G = C42.F5order 320 = 26·5

### 1st 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.F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — Dic5.D4 — C42.F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42.F5
 Upper central C1 — C2 — C22 — C2×C4 — C42

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, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, M4(2), C2×Q8, Dic5, C20, C2×C10, C4.10D4, C4⋊Q8, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C42.3C4, C4⋊Dic5, C4×C20, C22.F5, C2×Dic10, Dic5.D4, C202Q8, C42.F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C42.3C4, C22⋊F5, D10.D4, C42.F5

Character table of C42.F5

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 10A 10B 10C 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L size 1 1 2 4 4 4 20 20 40 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 -2 0 -2 2 0 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 -2 0 2 -2 0 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 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 ρ12 4 4 4 -4 4 -4 0 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 ρ13 4 4 4 4 4 4 0 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 -4 0 0 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 ρ15 4 4 4 0 -4 0 0 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 ρ16 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 ρ17 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 ρ18 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 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 2 0 -2 0 0 0 4 0 0 0 0 0 0 -4 2 2 0 -2 -2 -2 -2 0 0 0 2 2 symplectic lifted from C42.3C4, Schur index 2 ρ21 4 -4 0 -2 0 2 0 0 0 4 0 0 0 0 0 0 -4 -2 -2 0 2 2 2 2 0 0 0 -2 -2 symplectic lifted from C42.3C4, Schur index 2 ρ22 4 -4 0 2 0 -2 0 0 0 -1 0 0 0 0 -√5 √5 1 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 symplectic faithful, Schur index 2 ρ23 4 -4 0 -2 0 2 0 0 0 -1 0 0 0 0 -√5 √5 1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 2ζ4ζ54+2ζ4ζ52+ζ4 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 symplectic faithful, Schur index 2 ρ24 4 -4 0 2 0 -2 0 0 0 -1 0 0 0 0 -√5 √5 1 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ53+2ζ4ζ5+ζ4 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 symplectic faithful, Schur index 2 ρ25 4 -4 0 -2 0 2 0 0 0 -1 0 0 0 0 -√5 √5 1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 symplectic faithful, Schur index 2 ρ26 4 -4 0 -2 0 2 0 0 0 -1 0 0 0 0 √5 -√5 1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 symplectic faithful, Schur index 2 ρ27 4 -4 0 2 0 -2 0 0 0 -1 0 0 0 0 √5 -√5 1 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 2ζ43ζ54+2ζ43ζ53+ζ43 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 symplectic faithful, Schur index 2 ρ28 4 -4 0 2 0 -2 0 0 0 -1 0 0 0 0 √5 -√5 1 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 symplectic faithful, Schur index 2 ρ29 4 -4 0 -2 0 2 0 0 0 -1 0 0 0 0 √5 -√5 1 -ζ4ζ53+ζ4ζ52+ζ53+ζ52+1 ζ43ζ54-ζ43ζ5+ζ54+ζ5+1 2ζ43ζ54+2ζ43ζ53+ζ43 ζ4ζ53+ζ4ζ52+2ζ4ζ5+ζ4+ζ53+ζ52 ζ43ζ54+2ζ43ζ53+ζ43ζ5+ζ43+ζ54+ζ5 ζ43ζ54+2ζ43ζ52+ζ43ζ5+ζ43+ζ54+ζ5 2ζ4ζ54+ζ4ζ53+ζ4ζ52+ζ4+ζ53+ζ52 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 -ζ43ζ54+ζ43ζ5+ζ54+ζ5+1 ζ4ζ53-ζ4ζ52+ζ53+ζ52+1 symplectic faithful, Schur index 2

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

 1 0 0 0 0 1 0 0 0 0 30 32 0 0 9 11
,
 30 32 0 0 9 11 0 0 0 0 11 9 0 0 32 30
,
 34 40 0 0 1 0 0 0 0 0 7 7 0 0 34 40
,
 0 0 1 0 0 0 0 1 22 22 0 0 32 19 0 0
`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`

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