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G = C23⋊D10order 160 = 25·5

1st semidirect product of C23 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D105D4, C231D10, C52C22≀C2, (C2×D4)⋊3D5, (C2×C10)⋊2D4, (C2×C4)⋊2D10, (D4×C10)⋊8C2, C2.25(D4×D5), (C2×C20)⋊7C22, C10.49(C2×D4), (C23×D5)⋊2C2, C222(C5⋊D4), C23.D510C2, D10⋊C414C2, (C2×C10).52C23, (C22×C10)⋊3C22, (C2×Dic5)⋊2C22, C22.59(C22×D5), (C22×D5).28C22, (C2×C5⋊D4)⋊4C2, C2.13(C2×C5⋊D4), SmallGroup(160,158)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23⋊D10
C1C5C10C2×C10C22×D5C23×D5 — C23⋊D10
C5C2×C10 — C23⋊D10
C1C22C2×D4

Generators and relations for C23⋊D10
 G = < a,b,c,d,e | a2=b2=c2=d10=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 496 in 130 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×3], C22, C22 [×2], C22 [×21], C5, C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×3], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, Dic5 [×2], C20, D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×5], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, C23⋊D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4×D5 [×2], C2×C5⋊D4, C23⋊D10

Smallest permutation representation of C23⋊D10
On 40 points
Generators in S40
(1 32)(2 12)(3 34)(4 14)(5 36)(6 16)(7 38)(8 18)(9 40)(10 20)(11 30)(13 22)(15 24)(17 26)(19 28)(21 33)(23 35)(25 37)(27 39)(29 31)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 31)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 40)(18 39)(19 38)(20 37)

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

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

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

C23⋊D10 is a maximal subgroup of
C53C2≀C4  C23.3D20  C23⋊D20  2+ 1+42D5  C4212D10  D2023D4  C4216D10  C4217D10  D5×C22≀C2  C243D10  C244D10  C24.33D10  C24.34D10  C245D10  C10.372+ 1+4  C4⋊C421D10  C10.382+ 1+4  D2019D4  C10.402+ 1+4  D2020D4  C10.422+ 1+4  C10.462+ 1+4  C10.482+ 1+4  C10.1202+ 1+4  C4⋊C428D10  C10.612+ 1+4  C10.1222+ 1+4  C10.622+ 1+4  C10.682+ 1+4  C4220D10  C4221D10  C4222D10  C4226D10  D2011D4  C4228D10  D4×C5⋊D4  C248D10  (C2×C20)⋊15D4  C10.1452+ 1+4  C10.1462+ 1+4  D304D4  (C2×C30)⋊D4  D3019D4  D308D4  D3017D4  D10⋊S4
C23⋊D10 is a maximal quotient of
C24.46D10  C23⋊Dic10  C24.48D10  C24.12D10  C24.14D10  C232D20  (C2×C4)⋊Dic10  D105(C4⋊C4)  (C2×C4)⋊3D20  C242D10  D2016D4  D2017D4  Dic1017D4  D20.36D4  D20.37D4  Dic10.37D4  C22⋊C4⋊D10  C425D10  D20.14D4  D205D4  D20.15D4  D20⋊D4  Dic10⋊D4  D106SD16  D108SD16  D207D4  Dic10.16D4  D105Q16  D20.17D4  D2018D4  D20.38D4  D20.39D4  D20.40D4  C24.18D10  C24.21D10  D304D4  (C2×C30)⋊D4  D3019D4  D308D4  D3017D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C5A5B10A···10F10G···10N20A20B20C20D
order122222222224445510···1010···1020202020
size11112241010101042020222···24···44444

34 irreducible representations

dim1111112222224
type++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10C5⋊D4D4×D5
kernelC23⋊D10D10⋊C4C23.D5C2×C5⋊D4D4×C10C23×D5D10C2×C10C2×D4C2×C4C23C22C2
# reps1212114222484

Matrix representation of C23⋊D10 in GL4(𝔽41) generated by

17600
342400
0001
0010
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
0600
34700
00400
0001
,
34600
33700
0010
00040
G:=sub<GL(4,GF(41))| [17,34,0,0,6,24,0,0,0,0,0,1,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,34,0,0,6,7,0,0,0,0,40,0,0,0,0,1],[34,33,0,0,6,7,0,0,0,0,1,0,0,0,0,40] >;

C23⋊D10 in GAP, Magma, Sage, TeX

C_2^3\rtimes D_{10}
% in TeX

G:=Group("C2^3:D10");
// GroupNames label

G:=SmallGroup(160,158);
// by ID

G=gap.SmallGroup(160,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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