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G = D20⋊18D4  order 320 = 26·5

6th semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20⋊18D4, Dic10⋊18D4, M4(2)⋊5D10, (C5×D4)⋊7D4, (C5×Q8)⋊7D4, (C2×D4)⋊4D10, C8⋊C22⋊1D5, C20⋊D4⋊7C2, D4⋊4(C5⋊D4), C5⋊4(D4⋊4D4), C4○D4.5D10, Q8⋊4(C5⋊D4), C4.104(D4×D5), D20⋊7C4⋊9C2, D4⋊8D10⋊2C2, C20.194(C2×D4), (D4×C10)⋊4C22, (C22×D5).5D4, C22.35(D4×D5), C10.63C22≀C2, D4⋊2Dic5⋊6C2, D4.D10⋊5C2, C20.46D4⋊9C2, (C2×C20).13C23, (C4×Dic5)⋊5C22, C4.Dic5⋊8C22, C4○D20.23C22, C2.31(C23⋊D10), (C2×D20).132C22, (C5×M4(2))⋊15C22, (C5×C8⋊C22)⋊5C2, C4.50(C2×C5⋊D4), (C2×C10).34(C2×D4), (C2×C4).13(C22×D5), (C5×C4○D4).11C22, SmallGroup(320,825)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C20 — D20⋊18D4
Chief series
C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D4⋊8D10 — D20⋊18D4
Lower central
C5 — C10 — C2×C20 — D20⋊18D4
Upper central
C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D20⋊18D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a18b, dcd=c-1 >

Subgroups: 878 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], D5 [×3], C10, C10 [×3], C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4, C4○D4 [×3], Dic5 [×3], C20 [×2], C20, D10 [×7], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C4⋊1D4, C8⋊C22, C8⋊C22, 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5 [×3], D20, D20 [×4], C2×Dic5, C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×3], C5×Q8, C22×D5 [×2], C22×D5 [×2], C22×C10, D4⋊4D4, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, C5×M4(2), C5×D8, C5×SD16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5 [×3], Q8⋊2D5, C2×C5⋊D4 [×2], D4×C10, C5×C4○D4, C20.46D4, D20⋊7C4, D4⋊2Dic5, D4.D10, C20⋊D4, C5×C8⋊C22, D4⋊8D10, D20⋊18D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4⋊4D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20⋊18D4

Smallest permutation representation of D20⋊18D4
►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)

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

(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222222244444455881010101010···1020202020202040404040
size112482020202242020202284022448···84444888888

38 irreducible representations

dim11111111222222222224448
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10C5⋊D4C5⋊D4D4⋊4D4D4×D5D4×D5D20⋊18D4
kernelD20⋊18D4C20.46D4D20⋊7C4D4⋊2Dic5D4.D10C20⋊D4C5×C8⋊C22D4⋊8D10Dic10D20C5×D4C5×Q8C22×D5C8⋊C22M4(2)C2×D4C4○D4D4Q8C5C4C22C1
# reps11111111111122222442222

Matrix representation of D20⋊18D4 ►in GL8(𝔽41)

040000000
17000000
000400000
00170000
000004000
00001000
0000404012
0000104040
,
00010000
00100000
01000000
10000000
00000010
0000404012
00001000
000040011
,
400000000
71000000
00100000
0034400000
00000100
000040000
000000400
000001040
,
400000000
71000000
004000000
00710000
00000100
00001000
0000114039
00000001

G:=sub<GL(8,GF(41))| [0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,40,1,0,0,0,0,40,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,2,40],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,2,0,1],[40,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,39,1] >;
 

D20⋊18D4 in GAP, Magma, Sage, TeX 

D_{20}\rtimes_{18}D_4
 
% in TeX
 
G:=Group("D20:18D4");
 
// GroupNames label
 
G:=SmallGroup(320,825);
 
// by ID
 
G=gap.SmallGroup(320,825);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,570,1684,851,438,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^18*b,d*c*d=c^-1>;
 
// generators/relations
 

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