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G = C426F5order 320 = 26·5

3rd semidirect product of C42 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C426F5, C20.8C42, C4⋊F54C4, (C4×C20)⋊7C4, C4.F54C4, C4.1(C4×F5), D5.1C4≀C2, (C4×D5).86D4, C4.25(C4⋊F5), C20.25(C4⋊C4), (C4×D5).28Q8, C51(C426C4), (C4×Dic5)⋊24C4, D10.19(C4⋊C4), D5⋊M4(2).7C2, (D5×C42).23C2, Dic5.21(C4⋊C4), (C2×Dic5).253D4, (C22×D5).140D4, D10.28(C22⋊C4), C2.3(D10.3Q8), C22.12(C22⋊F5), C10.1(C2.C42), Dic5.28(C22⋊C4), D10.C23.7C2, (C4×D5).61(C2×C4), (C2×C4).115(C2×F5), (C2×C20).136(C2×C4), (C2×C4×D5).401C22, (C2×C10).12(C22⋊C4), SmallGroup(320,200)

Series: Derived Chief Lower central Upper central

C1C20 — C426F5
C1C5C10D10C22×D5C2×C4×D5D10.C23 — C426F5
C5C10C20 — C426F5
C1C4C2×C4C42

Generators and relations for C426F5
 G = < a,b,c,d | a4=b4=c5=d4=1, dad-1=ab=ba, ac=ca, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 450 in 110 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×2], D5, C10, C10, C42, C42 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×3], C22×C4 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2, C2×M4(2), C5⋊C8 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5 [×2], C22×D5, C426C4, C4×Dic5, C4×Dic5, C4×C20, D5⋊C8, C4.F5 [×2], C4×F5, C4⋊F5 [×2], C22.F5, C22⋊F5, C2×C4×D5, C2×C4×D5, D5×C42, D5⋊M4(2), D10.C23, C426F5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C4≀C2 [×2], C2×F5, C426C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C426F5

Smallest permutation representation of C426F5
On 40 points
Generators in S40
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(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)
(1 23)(2 25 5 21)(3 22 4 24)(6 28)(7 30 10 26)(8 27 9 29)(11 33)(12 35 15 31)(13 32 14 34)(16 38)(17 40 20 36)(18 37 19 39)

G:=sub<Sym(40)| (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,33)(12,35,15,31)(13,32,14,34)(16,38)(17,40,20,36)(18,37,19,39)>;

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (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), (1,23)(2,25,5,21)(3,22,4,24)(6,28)(7,30,10,26)(8,27,9,29)(11,33)(12,35,15,31)(13,32,14,34)(16,38)(17,40,20,36)(18,37,19,39) );

G=PermutationGroup([(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(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)], [(1,23),(2,25,5,21),(3,22,4,24),(6,28),(7,30,10,26),(8,27,9,29),(11,33),(12,35,15,31),(13,32,14,34),(16,38),(17,40,20,36),(18,37,19,39)])

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I4J···4N4O4P4Q4R 5 8A8B8C8D10A10B10C20A···20L
order122222444···4444···444445888810101020···20
size1125510112···25510···10202020204202020204444···4

44 irreducible representations

dim1111111122222444444
type+++++-+++++
imageC1C2C2C2C4C4C4C4D4Q8D4D4C4≀C2F5C2×F5C4×F5C4⋊F5C22⋊F5C426F5
kernelC426F5D5×C42D5⋊M4(2)D10.C23C4×Dic5C4×C20C4.F5C4⋊F5C4×D5C4×D5C2×Dic5C22×D5D5C42C2×C4C4C4C22C1
# reps1111224411118112228

Matrix representation of C426F5 in GL6(𝔽41)

4000000
090000
001000
000100
0000400
0000040
,
3200000
090000
0040000
0004000
0000400
0000040
,
100000
010000
00344000
001000
000077
00003440
,
090000
900000
000010
000001
001000
00344000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C426F5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6F_5
% in TeX

G:=Group("C4^2:6F5");
// GroupNames label

G:=SmallGroup(320,200);
// by ID

G=gap.SmallGroup(320,200);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,1684,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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