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G = C22⋊D20order 160 = 25·5

The semidirect product of C22 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D104D4, C222D20, C23.15D10, C51C22≀C2, (C2×C10)⋊1D4, (C2×C4)⋊1D10, C2.7(D4×D5), (C2×D20)⋊2C2, C22⋊C42D5, C2.7(C2×D20), C10.5(C2×D4), (C2×C20)⋊1C22, (C23×D5)⋊1C2, D10⋊C44C2, (C2×C10).23C23, (C2×Dic5)⋊1C22, (C22×D5)⋊1C22, C22.41(C22×D5), (C22×C10).12C22, (C2×C5⋊D4)⋊1C2, (C5×C22⋊C4)⋊3C2, SmallGroup(160,103)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C22⋊D20
C1C5C10C2×C10C22×D5C23×D5 — C22⋊D20
C5C2×C10 — C22⋊D20
C1C22C22⋊C4

Generators and relations for C22⋊D20
 G = < a,b,c,d | a2=b2=c20=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 544 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×3], C22, C22 [×2], C22 [×21], C5, C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], D5 [×5], C10, C10 [×2], C10 [×2], C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×2], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×D5 [×6], C22×C10, D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, C22⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, C2×D20, D4×D5 [×2], C22⋊D20

Smallest permutation representation of C22⋊D20
On 40 points
Generators in S40
(2 35)(4 37)(6 39)(8 21)(10 23)(12 25)(14 27)(16 29)(18 31)(20 33)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)

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

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

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

C22⋊D20 is a maximal subgroup of
C5⋊C2≀C4  C23⋊D20  C24.27D10  C233D20  C428D10  C429D10  C4210D10  C4212D10  D4×D20  D45D20  C4217D10  D5×C22≀C2  C244D10  C24.34D10  C10.372+ 1+4  C10.382+ 1+4  D2019D4  C10.482+ 1+4  C4⋊C426D10  D2021D4  C10.532+ 1+4  C10.562+ 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C4⋊C428D10  C10.612+ 1+4  C10.682+ 1+4  C4218D10  D2010D4  C4220D10  C4222D10  C4223D10  C4224D10  C4225D10  D64D20  D305D4  (C2×C6)⋊8D20  D3018D4  D3016D4  A4⋊D20
C22⋊D20 is a maximal quotient of
(C2×Dic5)⋊Q8  (C2×C4)⋊9D20  D103(C4⋊C4)  (C2×C20)⋊5D4  (C2×Dic5)⋊3D4  (C2×C4).20D20  D20.31D4  D2013D4  D20.32D4  D2014D4  Dic1014D4  C22⋊Dic20  C23⋊D20  C23.5D20  D20.1D4  D201D4  D20.4D4  D20.5D4  D4⋊D20  D20.8D4  D43D20  D4.D20  Q82D20  D104Q16  Q8.D20  D204D4  D44D20  M4(2)⋊D10  D4.9D20  D4.10D20  C24.47D10  C24.48D10  C23.45D20  C232D20  C24.16D10  D64D20  D305D4  (C2×C6)⋊8D20  D3018D4  D3016D4

34 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C5A5B10A···10F10G10H10I10J20A···20H
order122222222224445510···101010101020···20
size11112210101010204420222···244444···4

34 irreducible representations

dim1111112222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D20D4×D5
kernelC22⋊D20D10⋊C4C5×C22⋊C4C2×D20C2×C5⋊D4C23×D5D10C2×C10C22⋊C4C2×C4C23C22C2
# reps1212114224284

Matrix representation of C22⋊D20 in GL4(𝔽41) generated by

1000
0100
0010
002140
,
1000
0100
00400
00040
,
91100
301400
00137
00040
,
323000
11900
00404
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,37,40],[32,11,0,0,30,9,0,0,0,0,40,0,0,0,4,1] >;

C22⋊D20 in GAP, Magma, Sage, TeX

C_2^2\rtimes D_{20}
% in TeX

G:=Group("C2^2:D20");
// GroupNames label

G:=SmallGroup(160,103);
// by ID

G=gap.SmallGroup(160,103);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^20=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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