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G = C11⋊C40order 440 = 23·5·11

The semidirect product of C11 and C40 acting via C40/C4=C10

metacyclic, supersoluble, monomial, Z-group

Aliases: C11⋊C40, C22.C20, C4.2F11, C44.2C10, C11⋊C8⋊C5, C11⋊C5⋊C8, C2.(C11⋊C20), (C2×C11⋊C5).C4, (C4×C11⋊C5).2C2, SmallGroup(440,1)

Series: Derived Chief Lower central Upper central

C1C11 — C11⋊C40
C1C11C22C44C4×C11⋊C5 — C11⋊C40
C11 — C11⋊C40
C1C4

Generators and relations for C11⋊C40
 G = < a,b | a11=b40=1, bab-1=a8 >

11C5
11C10
11C8
11C20
11C40

Smallest permutation representation of C11⋊C40
On 88 points
Generators in S88
(1 17 72 41 33 9 56 80 88 25 64)(2 49 10 73 65 81 34 18 26 57 42)(3 27 82 11 43 19 66 50 58 35 74)(4 59 20 83 75 51 44 28 36 67 12)(5 37 52 21 13 29 76 60 68 45 84)(6 69 30 53 85 61 14 38 46 77 22)(7 47 62 31 23 39 86 70 78 15 54)(8 79 40 63 55 71 24 48 16 87 32)
(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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)

G:=sub<Sym(88)| (1,17,72,41,33,9,56,80,88,25,64)(2,49,10,73,65,81,34,18,26,57,42)(3,27,82,11,43,19,66,50,58,35,74)(4,59,20,83,75,51,44,28,36,67,12)(5,37,52,21,13,29,76,60,68,45,84)(6,69,30,53,85,61,14,38,46,77,22)(7,47,62,31,23,39,86,70,78,15,54)(8,79,40,63,55,71,24,48,16,87,32), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)>;

G:=Group( (1,17,72,41,33,9,56,80,88,25,64)(2,49,10,73,65,81,34,18,26,57,42)(3,27,82,11,43,19,66,50,58,35,74)(4,59,20,83,75,51,44,28,36,67,12)(5,37,52,21,13,29,76,60,68,45,84)(6,69,30,53,85,61,14,38,46,77,22)(7,47,62,31,23,39,86,70,78,15,54)(8,79,40,63,55,71,24,48,16,87,32), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88) );

G=PermutationGroup([(1,17,72,41,33,9,56,80,88,25,64),(2,49,10,73,65,81,34,18,26,57,42),(3,27,82,11,43,19,66,50,58,35,74),(4,59,20,83,75,51,44,28,36,67,12),(5,37,52,21,13,29,76,60,68,45,84),(6,69,30,53,85,61,14,38,46,77,22),(7,47,62,31,23,39,86,70,78,15,54),(8,79,40,63,55,71,24,48,16,87,32)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)])

44 conjugacy classes

class 1  2 4A4B5A5B5C5D8A8B8C8D10A10B10C10D 11 20A···20H 22 40A···40P44A44B
order124455558888101010101120···202240···404444
size11111111111111111111111111111011···111011···111010

44 irreducible representations

dim11111111101010
type+++-
imageC1C2C4C5C8C10C20C40F11C11⋊C20C11⋊C40
kernelC11⋊C40C4×C11⋊C5C2×C11⋊C5C11⋊C8C11⋊C5C44C22C11C4C2C1
# reps112444816112

Matrix representation of C11⋊C40 in GL10(𝔽881)

880100000000
880010000000
880001000000
880000100000
880000010000
880000001000
880000000100
880000000010
880000000001
880000000000
,
0560321321321560081400
3210032125456005600560
3218140321056056003210
3215600032100560321814
0032132100814560321560
3215603212540056056000
2545603210056000321560
0560032132105600254560
3210321032181456000560
0025403215605605603210

G:=sub<GL(10,GF(881))| [880,880,880,880,880,880,880,880,880,880,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[0,321,321,321,0,321,254,0,321,0,560,0,814,560,0,560,560,560,0,0,321,0,0,0,321,321,321,0,321,254,321,321,321,0,321,254,0,321,0,0,321,254,0,321,0,0,0,321,321,321,560,560,560,0,0,0,560,0,814,560,0,0,560,0,814,560,0,560,560,560,814,560,0,560,560,560,0,0,0,560,0,0,321,321,321,0,321,254,0,321,0,560,0,814,560,0,560,560,560,0] >;

C11⋊C40 in GAP, Magma, Sage, TeX

C_{11}\rtimes C_{40}
% in TeX

G:=Group("C11:C40");
// GroupNames label

G:=SmallGroup(440,1);
// by ID

G=gap.SmallGroup(440,1);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-11,50,42,10004,4509]);
// Polycyclic

G:=Group<a,b|a^11=b^40=1,b*a*b^-1=a^8>;
// generators/relations

Export

Subgroup lattice of C11⋊C40 in TeX

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