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## G = D4×C11⋊C5order 440 = 23·5·11

### Direct product of D4 and C11⋊C5

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

 Derived series C1 — C22 — D4×C11⋊C5
 Chief series C1 — C11 — C22 — C2×C11⋊C5 — C22×C11⋊C5 — D4×C11⋊C5
 Lower central C11 — C22 — D4×C11⋊C5
 Upper central C1 — C2 — D4

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 >

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 2A 2B 2C 4 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10L 11A 11B 20A 20B 20C 20D 22A 22B 22C 22D 22E 22F 44A 44B order 1 2 2 2 4 5 5 5 5 10 10 10 10 10 ··· 10 11 11 20 20 20 20 22 22 22 22 22 22 44 44 size 1 1 2 2 2 11 11 11 11 11 11 11 11 22 ··· 22 5 5 22 22 22 22 5 5 10 10 10 10 10 10

35 irreducible representations

 dim 1 1 1 1 1 1 10 2 2 5 5 5 type + + + + image C1 C2 C2 C5 C10 C10 D4×C11⋊C5 D4 C5×D4 C11⋊C5 C2×C11⋊C5 C2×C11⋊C5 kernel D4×C11⋊C5 C4×C11⋊C5 C22×C11⋊C5 D4×C11 C44 C2×C22 C1 C11⋊C5 C11 D4 C4 C22 # reps 1 1 2 4 4 8 2 1 4 2 2 4

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

 660 104 0 0 0 0 0 572 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 0 0 0 0 0 0 89 660 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 1 660 45 1 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
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 616 659 1 615 43 0 0 615 43 1 659 44 0 0 0 1 0 0 0

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

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