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G = D4×D11order 176 = 24·11

Direct product of D4 and D11

Aliases: D4×D11, C41D22, C44⋊C22, D443C2, C221D22, D222C22, C22.5C23, Dic111C22, C112(C2×D4), (C2×C22)⋊C22, (C4×D11)⋊1C2, (D4×C11)⋊2C2, C11⋊D41C2, (C22×D11)⋊2C2, C2.6(C22×D11), SmallGroup(176,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — D4×D11
 Chief series C1 — C11 — C22 — D22 — C22×D11 — D4×D11
 Lower central C11 — C22 — D4×D11
 Upper central C1 — C2 — D4

Generators and relations for D4×D11
G = < a,b,c,d | a4=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 320 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C11, C2×D4, D11, D11, C22, C22, Dic11, C44, D22, D22, D22, C2×C22, C4×D11, D44, C11⋊D4, D4×C11, C22×D11, D4×D11
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, D22, C22×D11, D4×D11

Smallest permutation representation of D4×D11
On 44 points
Generators in S44
(1 32 21 43)(2 33 22 44)(3 23 12 34)(4 24 13 35)(5 25 14 36)(6 26 15 37)(7 27 16 38)(8 28 17 39)(9 29 18 40)(10 30 19 41)(11 31 20 42)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)

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

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

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

D4×D11 is a maximal subgroup of
D4⋊D22  D88⋊C2  D46D22  D48D22
D4×D11 is a maximal quotient of
C22⋊Dic22  Dic114D4  C22⋊D44  D22.D4  D22⋊D4  Dic11.D4  C44⋊Q8  D44⋊C4  D22.5D4  C42D44  D22⋊Q8  D4⋊D22  D83D11  D88⋊C2  D4.D22  Q8.D22  Q16⋊D11  D885C2  C23⋊D22  C442D4  Dic11⋊D4  C44⋊D4

35 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 11A ··· 11E 22A ··· 22E 22F ··· 22O 44A ··· 44E order 1 2 2 2 2 2 2 2 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 2 11 11 22 22 2 22 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D11 D22 D22 D4×D11 kernel D4×D11 C4×D11 D44 C11⋊D4 D4×C11 C22×D11 D11 D4 C4 C22 C1 # reps 1 1 1 2 1 2 2 5 5 10 5

Matrix representation of D4×D11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 53 4 0 0 54 36
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 18 88
,
 7 1 0 0 88 0 0 0 0 0 1 0 0 0 0 1
,
 27 41 0 0 30 62 0 0 0 0 88 0 0 0 0 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,53,54,0,0,4,36],[1,0,0,0,0,1,0,0,0,0,1,18,0,0,0,88],[7,88,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[27,30,0,0,41,62,0,0,0,0,88,0,0,0,0,88] >;

D4×D11 in GAP, Magma, Sage, TeX

D_4\times D_{11}
% in TeX

G:=Group("D4xD11");
// GroupNames label

G:=SmallGroup(176,31);
// by ID

G=gap.SmallGroup(176,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,97,4004]);
// Polycyclic

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

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