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G = C53C2≀C4order 320 = 26·5

The semidirect product of C5 and C2≀C4 acting via C2≀C4/C23⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C53C2≀C4, C23⋊C41D5, (C2×C4).1D20, (C2×C20).1D4, (C2×D4).1D10, (C23×D5)⋊1C4, C23.D51C4, C23.1(C4×D5), C20.D41C2, (C22×C10).8D4, C23⋊D10.1C2, (D4×C10).1C22, C23.1(C5⋊D4), C10.28(C23⋊C4), C22.8(D10⋊C4), C2.8(C23.1D10), (C5×C23⋊C4)⋊1C2, (C22×C10).1(C2×C4), (C2×C10).65(C22⋊C4), SmallGroup(320,29)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C53C2≀C4
C1C5C10C2×C10C22×C10D4×C10C23⋊D10 — C53C2≀C4
C5C10C2×C10C22×C10 — C53C2≀C4
C1C2C22C2×D4C23⋊C4

Generators and relations for C53C2≀C4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 574 in 94 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], D5 [×2], C10, C10 [×3], C22⋊C4 [×3], M4(2), C2×D4, C2×D4, C24, Dic5, C20 [×2], D10 [×8], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22≀C2, C52C8, C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C5×D4, C22×D5 [×4], C22×C10 [×2], C2≀C4, C4.Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C5⋊D4, D4×C10, C23×D5, C20.D4, C5×C23⋊C4, C23⋊D10, C53C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C2≀C4, D10⋊C4, C23.1D10, C53C2≀C4

Smallest permutation representation of C53C2≀C4
On 40 points
Generators in S40
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 32)(22 31)(23 35)(24 34)(25 33)(26 37)(27 36)(28 40)(29 39)(30 38)
(1 14)(2 15)(3 11)(4 12)(5 13)(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)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D5A5B8A8B10A10B10C···10H10I10J20A···20J
order122222244445588101010···10101020···20
size11244202048840224040224···4888···8

35 irreducible representations

dim11111122222224448
type++++++++++++
imageC1C2C2C2C4C4D4D4D5D10D20C4×D5C5⋊D4C23⋊C4C2≀C4C23.1D10C53C2≀C4
kernelC53C2≀C4C20.D4C5×C23⋊C4C23⋊D10C23.D5C23×D5C2×C20C22×C10C23⋊C4C2×D4C2×C4C23C23C10C5C2C1
# reps11112211224441242

Matrix representation of C53C2≀C4 in GL6(𝔽41)

35360000
40400000
001000
000100
000010
000001
,
660000
1350000
0040000
0004000
0001812
0000040
,
100000
010000
00401800
000100
000234039
000001
,
100000
010000
001000
000100
0000400
00023040
,
100000
010000
0040000
0004000
0000400
0000040
,
900000
090000
0000400
0001032
0040000
00123040

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,36,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,18,0,0,0,0,0,1,0,0,0,0,0,2,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,18,1,23,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,1,0,0,0,1,0,23,0,0,40,0,0,0,0,0,0,32,0,40] >;

C53C2≀C4 in GAP, Magma, Sage, TeX

C_5\rtimes_3C_2\wr C_4
% in TeX

G:=Group("C5:3C2wrC4");
// GroupNames label

G:=SmallGroup(320,29);
// by ID

G=gap.SmallGroup(320,29);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,346,297,851,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations

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