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G = C23⋊D20  order 320 = 26·5

The semidirect product of C23 and D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊D20, (C2×C4)⋊D20, C5⋊1C2≀C22, (C2×C20)⋊1D4, C23⋊C4⋊3D5, C22⋊C4⋊1D10, (C2×Dic5)⋊1D4, (C22×D5)⋊1D4, (C22×C10)⋊2D4, C23⋊D10⋊1C2, D4⋊6D10⋊1C2, C22⋊D20⋊1C2, (C2×D4).11D10, C22.8(C2×D20), C22.24(D4×D5), C10.13C22≀C2, (D4×C10).8C22, (C23×D5)⋊1C22, C23.D5⋊1C22, C23.2(C22×D5), C23.1D10⋊1C2, (C22×C10).2C23, C2.16(C22⋊D20), (C5×C23⋊C4)⋊4C2, (C2×C10).17(C2×D4), (C5×C22⋊C4)⋊1C22, (C2×C5⋊D4).2C22, SmallGroup(320,368)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — C23⋊D20
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C2×C5⋊D4 — D4⋊6D10 — C23⋊D20
Lower central
C5 — C10 — C22×C10 — C23⋊D20
Upper central
C1 — C2 — C23 — C23⋊C4

Generators and relations for C23⋊D20
 G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1150 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×8], C4 [×6], C22, C22 [×2], C22 [×18], C5, C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], D5 [×4], C10, C10 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic5 [×3], C20 [×3], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4, C23⋊C4 [×2], C22≀C2 [×3], 2+ 1+4, Dic10, C4×D5 [×2], D20 [×5], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10 [×2], C2≀C22, D10⋊C4 [×3], C23.D5, C5×C22⋊C4 [×2], C2×D20 [×2], C4○D20, D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×2], D4×C10, C23×D5, C23.1D10 [×2], C5×C23⋊C4, C22⋊D20 [×2], C23⋊D10, D4⋊6D10, C23⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, C2≀C22, C2×D20, D4×D5 [×2], C22⋊D20, C23⋊D20

Smallest permutation representation of C23⋊D20
►On 40 points
Generators in S40
(1 24)(2 40)(3 36)(4 32)(5 28)(6 38)(7 34)(8 30)(9 26)(10 22)(11 29)(12 35)(13 21)(14 27)(15 33)(16 39)(17 25)(18 31)(19 37)(20 23)

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F5A5B10A10B10C···10H10I10J20A···20J
order122222222244444455101010···10101020···20
size1122242020202048820204022224···4888···8

38 irreducible representations

dim111111222222222448
type++++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D4D5D10D10D20D20C2≀C22D4×D5C23⋊D20
kernelC23⋊D20C23.1D10C5×C23⋊C4C22⋊D20C23⋊D10D4⋊6D10C2×Dic5C2×C20C22×D5C22×C10C23⋊C4C22⋊C4C2×D4C2×C4C23C5C22C1
# reps121211212124244242

Matrix representation of C23⋊D20 ►in GL8(𝔽41)

1038230000
0136230000
004000000
000400000
000000400
000000040
000040000
000004000
,
400000000
040000000
004000000
000400000
00000100
00001000
00000001
00000010
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
01000000
4034000000
40331400000
763660000
000040000
00000100
00000001
000000400
,
01000000
10000000
140100000
343936400000
000040000
000004000
00000001
00000010

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,38,36,40,0,0,0,0,0,23,23,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,40,40,7,0,0,0,0,1,34,33,6,0,0,0,0,0,0,1,36,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,1,1,34,0,0,0,0,1,0,40,39,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C23⋊D20 in GAP, Magma, Sage, TeX 

C_2^3\rtimes D_{20}
 
% in TeX
 
G:=Group("C2^3:D20");
 
// GroupNames label
 
G:=SmallGroup(320,368);
 
// by ID
 
G=gap.SmallGroup(320,368);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,570,1684,438,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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