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G = D4×C11⋊C5order 440 = 23·5·11

Direct product of D4 and C11⋊C5

direct product, metacyclic, supersoluble, monomial

Aliases: D4×C11⋊C5, C443C10, (D4×C11)⋊C5, C113(C5×D4), (C2×C22)⋊3C10, C22.7(C2×C10), C4⋊(C2×C11⋊C5), C22⋊(C2×C11⋊C5), (C4×C11⋊C5)⋊3C2, (C22×C11⋊C5)⋊3C2, C2.2(C22×C11⋊C5), (C2×C11⋊C5).7C22, SmallGroup(440,13)

Series: Derived Chief Lower central Upper central

C1C22 — D4×C11⋊C5
C1C11C22C2×C11⋊C5C22×C11⋊C5 — D4×C11⋊C5
C11C22 — D4×C11⋊C5
C1C2D4

Generators and relations for D4×C11⋊C5
 G = < a,b,c,d | a4=b2=c11=d5=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

2C2
2C2
11C5
11C10
22C10
22C10
2C22
2C22
11C2×C10
11C2×C10
11C20
2C2×C11⋊C5
2C2×C11⋊C5
11C5×D4

Smallest permutation representation of D4×C11⋊C5
On 44 points
Generators in S44
(1 23 12 34)(2 24 13 35)(3 25 14 36)(4 26 15 37)(5 27 16 38)(6 28 17 39)(7 29 18 40)(8 30 19 41)(9 31 20 42)(10 32 21 43)(11 33 22 44)
(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)

G:=sub<Sym(44)| (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)>;

G:=Group( (1,23,12,34)(2,24,13,35)(3,25,14,36)(4,26,15,37)(5,27,16,38)(6,28,17,39)(7,29,18,40)(8,30,19,41)(9,31,20,42)(10,32,21,43)(11,33,22,44), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (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), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40) );

G=PermutationGroup([[(1,23,12,34),(2,24,13,35),(3,25,14,36),(4,26,15,37),(5,27,16,38),(6,28,17,39),(7,29,18,40),(8,30,19,41),(9,31,20,42),(10,32,21,43),(11,33,22,44)], [(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44)], [(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)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40)]])

35 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D10A10B10C10D10E···10L11A11B20A20B20C20D22A22B22C22D22E22F44A44B
order1222455551010101010···101111202020202222222222224444
size11222111111111111111122···22552222222255101010101010

35 irreducible representations

dim1111111022555
type++++
imageC1C2C2C5C10C10D4×C11⋊C5D4C5×D4C11⋊C5C2×C11⋊C5C2×C11⋊C5
kernelD4×C11⋊C5C4×C11⋊C5C22×C11⋊C5D4×C11C44C2×C22C1C11⋊C5C11D4C4C22
# reps112448214224

Matrix representation of D4×C11⋊C5 in GL7(𝔽661)

66010400000
572100000
0010000
0001000
0000100
0000010
0000001
,
1000000
8966000000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
00441660451
0010000
0001000
0000100
0000010
,
1000000
0100000
0010000
0000010
00616659161543
0061543165944
0001000

G:=sub<GL(7,GF(661))| [660,572,0,0,0,0,0,104,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,89,0,0,0,0,0,0,660,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,0,1,0,1,0,0,0,0,660,0,0,1,0,0,0,45,0,0,0,1,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,616,615,0,0,0,0,0,659,43,1,0,0,0,0,1,1,0,0,0,0,1,615,659,0,0,0,0,0,43,44,0] >;

D4×C11⋊C5 in GAP, Magma, Sage, TeX

D_4\times C_{11}\rtimes C_5
% in TeX

G:=Group("D4xC11:C5");
// GroupNames label

G:=SmallGroup(440,13);
// by ID

G=gap.SmallGroup(440,13);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-11,221,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^11=d^5=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Subgroup lattice of D4×C11⋊C5 in TeX

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