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G = C425F5order 320 = 26·5

2nd semidirect product of C42 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C425F5, (C4×C20)⋊4C4, C5⋊(C425C4), (C4×Dic5)⋊22C4, (D5×C42).17C2, D10.24(C4○D4), D10.3Q8.5C2, C10.8(C42⋊C2), (C22×F5).3C22, C22.68(C22×F5), D5.1(C422C2), (C22×D5).267C23, C2.11(D10.C23), (C2×C4).102(C2×F5), (C2×C20).102(C2×C4), (C2×C4×D5).362C22, (C2×C10).28(C22×C4), (C2×Dic5).176(C2×C4), SmallGroup(320,1028)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C425F5
C1C5D5D10C22×D5C22×F5D10.3Q8 — C425F5
C5C2×C10 — C425F5
C1C22C42

Generators and relations for C425F5
 G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 570 in 138 conjugacy classes, 50 normal (9 characteristic)
C1, C2 [×3], C2 [×4], C4 [×10], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×21], C23, D5 [×4], C10 [×3], C42, C42 [×3], C22×C4 [×7], Dic5 [×3], C20 [×3], F5 [×4], D10 [×6], C2×C10, C2.C42 [×6], C2×C42, C4×D5 [×6], C2×Dic5 [×3], C2×C20 [×3], C2×F5 [×12], C22×D5, C425C4, C4×Dic5 [×3], C4×C20, C2×C4×D5 [×3], C22×F5 [×4], D10.3Q8 [×6], D5×C42, C425F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×6], F5, C42⋊C2 [×3], C422C2 [×4], C2×F5 [×3], C425C4, C22×F5, D10.C23 [×3], C425F5

Smallest permutation representation of C425F5
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)
(1 61 6 66)(2 63 10 69)(3 65 9 67)(4 62 8 70)(5 64 7 68)(11 71 16 76)(12 73 20 79)(13 75 19 77)(14 72 18 80)(15 74 17 78)(21 51 26 56)(22 53 30 59)(23 55 29 57)(24 52 28 60)(25 54 27 58)(31 41 36 46)(32 43 40 49)(33 45 39 47)(34 42 38 50)(35 44 37 48)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L4M···4T 5 10A10B10C20A···20L
order122222224···44···44···4510101020···20
size111155552···210···1020···2044444···4

44 irreducible representations

dim111112444
type+++++
imageC1C2C2C4C4C4○D4F5C2×F5D10.C23
kernelC425F5D10.3Q8D5×C42C4×Dic5C4×C20D10C42C2×C4C2
# reps16162121312

Matrix representation of C425F5 in GL6(𝔽41)

40390000
110000
009000
000900
000090
000009
,
3200000
0320000
0022033
003819380
000381938
0033022
,
100000
010000
0040404040
001000
000100
000010
,
40390000
010000
003402727
002727034
00147140
00734347

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,34,27,14,7,0,0,0,27,7,34,0,0,27,0,14,34,0,0,27,34,0,7] >;

C425F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_5F_5
% in TeX

G:=Group("C4^2:5F5");
// GroupNames label

G:=SmallGroup(320,1028);
// by ID

G=gap.SmallGroup(320,1028);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,184,6278,1595]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;
// generators/relations

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