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G = C20⋊D6order 240 = 24·3·5

2nd semidirect product of C20 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202D6, D102D6, D204S3, D62D10, D151D4, D124D5, C122D10, C604C22, C30.14C23, Dic156C22, D30.13C22, C32(D4×D5), C52(S3×D4), C43(S3×D5), C153(C2×D4), (C5×D12)⋊6C2, (C3×D20)⋊6C2, (C4×D15)⋊7C2, C15⋊D43C2, (C6×D5)⋊2C22, (S3×C10)⋊2C22, C6.14(C22×D5), C10.14(C22×S3), (C2×S3×D5)⋊3C2, C2.17(C2×S3×D5), SmallGroup(240,138)

Series: Derived Chief Lower central Upper central

C1C30 — C20⋊D6
C1C5C15C30C6×D5C2×S3×D5 — C20⋊D6
C15C30 — C20⋊D6
C1C2C4

Generators and relations for C20⋊D6
 G = < a,b,c | a20=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 584 in 108 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6 [×2], D6 [×5], C2×C6 [×2], C15, C2×D4, Dic5, C20, D10 [×2], D10 [×5], C2×C10 [×2], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5 [×2], S3×D4, Dic15, C60, S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30, D4×D5, C15⋊D4 [×2], C3×D20, C5×D12, C4×D15, C2×S3×D5 [×2], C20⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C22×D5, S3×D4, S3×D5, D4×D5, C2×S3×D5, C20⋊D6

Smallest permutation representation of C20⋊D6
On 60 points
Generators in S60
(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)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 45 33 6 60 38)(2 44 34 5 41 37)(3 43 35 4 42 36)(7 59 39 20 46 32)(8 58 40 19 47 31)(9 57 21 18 48 30)(10 56 22 17 49 29)(11 55 23 16 50 28)(12 54 24 15 51 27)(13 53 25 14 52 26)
(1 23)(2 32)(3 21)(4 30)(5 39)(6 28)(7 37)(8 26)(9 35)(10 24)(11 33)(12 22)(13 31)(14 40)(15 29)(16 38)(17 27)(18 36)(19 25)(20 34)(41 59)(42 48)(43 57)(44 46)(45 55)(47 53)(49 51)(50 60)(52 58)(54 56)

G:=sub<Sym(60)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,33,6,60,38)(2,44,34,5,41,37)(3,43,35,4,42,36)(7,59,39,20,46,32)(8,58,40,19,47,31)(9,57,21,18,48,30)(10,56,22,17,49,29)(11,55,23,16,50,28)(12,54,24,15,51,27)(13,53,25,14,52,26), (1,23)(2,32)(3,21)(4,30)(5,39)(6,28)(7,37)(8,26)(9,35)(10,24)(11,33)(12,22)(13,31)(14,40)(15,29)(16,38)(17,27)(18,36)(19,25)(20,34)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56)>;

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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,33,6,60,38)(2,44,34,5,41,37)(3,43,35,4,42,36)(7,59,39,20,46,32)(8,58,40,19,47,31)(9,57,21,18,48,30)(10,56,22,17,49,29)(11,55,23,16,50,28)(12,54,24,15,51,27)(13,53,25,14,52,26), (1,23)(2,32)(3,21)(4,30)(5,39)(6,28)(7,37)(8,26)(9,35)(10,24)(11,33)(12,22)(13,31)(14,40)(15,29)(16,38)(17,27)(18,36)(19,25)(20,34)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56) );

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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,45,33,6,60,38),(2,44,34,5,41,37),(3,43,35,4,42,36),(7,59,39,20,46,32),(8,58,40,19,47,31),(9,57,21,18,48,30),(10,56,22,17,49,29),(11,55,23,16,50,28),(12,54,24,15,51,27),(13,53,25,14,52,26)], [(1,23),(2,32),(3,21),(4,30),(5,39),(6,28),(7,37),(8,26),(9,35),(10,24),(11,33),(12,22),(13,31),(14,40),(15,29),(16,38),(17,27),(18,36),(19,25),(20,34),(41,59),(42,48),(43,57),(44,46),(45,55),(47,53),(49,51),(50,60),(52,58),(54,56)])

C20⋊D6 is a maximal subgroup of
C4014D6  C405D6  D246D5  C408D6  D15⋊D8  D30.8D4  D15⋊SD16  D60⋊C22  D2024D6  D2025D6  D2026D6  S3×D4×D5  D2013D6  D2016D6  D2017D6
C20⋊D6 is a maximal quotient of
C4014D6  C405D6  D246D5  C408D6  Dic10.D6  Dic6.D10  D405S3  D30.3D4  D245D5  D30.4D4  D30.D4  Dic15⋊D4  Dic15.D4  D208Dic3  Dic158D4  D64Dic10  D30.2Q8  D30.7D4  Dic159D4  Dic152D4  D3012D4  C6010D4  Dic15.31D4  C122D20  C202D12  C20⋊Dic6  D30.27D4  D64D20  D304D4

33 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A10B10C10D10E10F 12 15A15B20A20B30A30B60A60B60C60D
order12222222344556661010101010101215152020303060606060
size11661010151522302222020221212121244444444444

33 irreducible representations

dim111111222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D10D10S3×D4S3×D5D4×D5C2×S3×D5C20⋊D6
kernelC20⋊D6C15⋊D4C3×D20C5×D12C4×D15C2×S3×D5D20D15D12C20D10C12D6C5C4C3C2C1
# reps121112122122412224

Matrix representation of C20⋊D6 in GL6(𝔽61)

6000000
0600000
000100
00604400
0000603
0000401
,
110000
6000000
00441700
0011700
0000158
0000060
,
60600000
010000
00174400
00604400
0000600
0000060

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,44,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,60,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,58,60],[60,0,0,0,0,0,60,1,0,0,0,0,0,0,17,60,0,0,0,0,44,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C20⋊D6 in GAP, Magma, Sage, TeX

C_{20}\rtimes D_6
% in TeX

G:=Group("C20:D6");
// GroupNames label

G:=SmallGroup(240,138);
// by ID

G=gap.SmallGroup(240,138);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^20=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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