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G = 2+ 1+4⋊D5order 320 = 26·5

1st semidirect product of 2+ 1+4 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2+ 1+41D5, (C5×D4)⋊15D4, (C5×Q8)⋊15D4, C20⋊D48C2, C55(D44D4), D47(C5⋊D4), C4○D4.9D10, Q87(C5⋊D4), D4⋊D105C2, (C2×D4).82D10, C20.217(C2×D4), C10.80C22≀C2, C20.D411C2, (C2×D20)⋊15C22, (C2×C20).21C23, (C4×Dic5)⋊8C22, (C22×C10).24D4, D42Dic511C2, C23.12(C5⋊D4), C4.Dic510C22, (C5×2+ 1+4)⋊1C2, (D4×C10).107C22, C2.14(C242D5), C4.64(C2×C5⋊D4), (C2×C10).42(C2×D4), (C2×C4).21(C22×D5), C22.14(C2×C5⋊D4), (C5×C4○D4).19C22, SmallGroup(320,868)

Series: Derived Chief Lower central Upper central

C1C2×C20 — 2+ 1+4⋊D5
C1C5C10C2×C10C2×C20C2×D20D4⋊D10 — 2+ 1+4⋊D5
C5C10C2×C20 — 2+ 1+4⋊D5
C1C2C2×C42+ 1+4

Generators and relations for 2+ 1+4⋊D5
 G = < a,b,c,d,e,f | a4=b2=d2=e5=f2=1, c2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a2c, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 670 in 168 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×3], D5, C10, C10 [×5], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×2], C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×8], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×D4 [×8], C5×Q8 [×2], C22×D5, C22×C10 [×2], C22×C10 [×2], D44D4, C4.Dic5 [×2], C4×Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C2×C5⋊D4 [×2], D4×C10, D4×C10 [×4], C5×C4○D4 [×2], C5×C4○D4 [×2], C20.D4, D42Dic5 [×2], C20⋊D4, D4⋊D10 [×2], C5×2+ 1+4, 2+ 1+4⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×6], C22×D5, D44D4, C2×C5⋊D4 [×3], C242D5, 2+ 1+4⋊D5

Smallest permutation representation of 2+ 1+4⋊D5
On 40 points
Generators in S40
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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 8)(2 7)(3 6)(4 10)(5 9)(12 15)(13 14)(17 20)(18 19)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A10B10C···10T20A···20L
order122222224444445588101010···1020···20
size11244444022442020224040224···44···4

50 irreducible representations

dim11111122222222248
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10C5⋊D4C5⋊D4C5⋊D4D44D42+ 1+4⋊D5
kernel2+ 1+4⋊D5C20.D4D42Dic5C20⋊D4D4⋊D10C5×2+ 1+4C5×D4C5×Q8C22×C102+ 1+4C2×D4C4○D4D4Q8C23C5C1
# reps11212122222488822

Matrix representation of 2+ 1+4⋊D5 in GL6(𝔽41)

4000000
0400000
0004000
001000
0000040
000010
,
4000000
010000
0000040
0000400
0004000
0040000
,
100000
010000
000100
0040000
0000040
000010
,
4000000
0400000
0000040
000010
000100
0040000
,
1000000
0370000
001000
000100
000010
000001
,
0180000
1600000
001000
0004000
000001
000010

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40,0,0,0],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

2+ 1+4⋊D5 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes D_5
% in TeX

G:=Group("ES+(2,2):D5");
// GroupNames label

G:=SmallGroup(320,868);
// by ID

G=gap.SmallGroup(320,868);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,570,1684,851,438,102,12550]);
// Polycyclic

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

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