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## G = D4⋊D22order 352 = 25·11

### 2nd semidirect product of D4 and D22 acting via D22/D11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D4⋊D22
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D4×D11 — D4⋊D22
 Lower central C11 — C22 — C44 — D4⋊D22
 Upper central C1 — C2 — C4 — D8

Generators and relations for D4⋊D22
G = < a,b,c,d | a8=b2=c11=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 506 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C11, M4(2), D8, D8, SD16, C2×D4, C4○D4, D11, C22, C22, C8⋊C22, Dic11, Dic11, C44, D22, D22, C2×C22, C11⋊C8, C88, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, D4×C11, C22×D11, C88⋊C2, C8⋊D11, D4⋊D11, D4.D11, C11×D8, D4×D11, D42D11, D4⋊D22
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C8⋊C22, D22, C22×D11, D4×D11, D4⋊D22

Smallest permutation representation of D4⋊D22
On 88 points
Generators in S88
(1 54 32 87 21 65 43 76)(2 55 33 88 22 66 44 77)(3 45 23 78 12 56 34 67)(4 46 24 79 13 57 35 68)(5 47 25 80 14 58 36 69)(6 48 26 81 15 59 37 70)(7 49 27 82 16 60 38 71)(8 50 28 83 17 61 39 72)(9 51 29 84 18 62 40 73)(10 52 30 85 19 63 41 74)(11 53 31 86 20 64 42 75)
(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(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)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 11A ··· 11E 22A ··· 22E 22F ··· 22O 44A ··· 44E 88A ··· 88J order 1 2 2 2 2 2 4 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 88 ··· 88 size 1 1 4 4 22 44 2 22 44 4 44 2 ··· 2 2 ··· 2 8 ··· 8 4 ··· 4 4 ··· 4

46 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D11 D22 D22 C8⋊C22 D4×D11 D4⋊D22 kernel D4⋊D22 C88⋊C2 C8⋊D11 D4⋊D11 D4.D11 C11×D8 D4×D11 D4⋊2D11 Dic11 D22 D8 C8 D4 C11 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 5 5 10 1 5 10

Matrix representation of D4⋊D22 in GL4(𝔽89) generated by

 0 0 15 35 0 0 67 59 14 46 18 12 55 82 77 71
,
 1 0 0 0 0 1 0 0 67 57 88 0 1 70 0 88
,
 11 1 0 0 23 75 0 0 0 0 0 1 0 0 88 86
,
 69 81 0 0 61 20 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(89))| [0,0,14,55,0,0,46,82,15,67,18,77,35,59,12,71],[1,0,67,1,0,1,57,70,0,0,88,0,0,0,0,88],[11,23,0,0,1,75,0,0,0,0,0,88,0,0,1,86],[69,61,0,0,81,20,0,0,0,0,0,1,0,0,1,0] >;

D4⋊D22 in GAP, Magma, Sage, TeX

D_4\rtimes D_{22}
% in TeX

G:=Group("D4:D22");
// GroupNames label

G:=SmallGroup(352,106);
// by ID

G=gap.SmallGroup(352,106);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,116,297,159,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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