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G = D42D11order 176 = 24·11

The semidirect product of D4 and D11 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D11, C4.5D22, Dic223C2, C22.6C23, C44.5C22, C22.1D22, D22.2C22, Dic11.4C22, (D4×C11)⋊3C2, (C4×D11)⋊2C2, C112(C4○D4), C11⋊D42C2, (C2×C22).C22, (C2×Dic11)⋊3C2, C2.7(C22×D11), SmallGroup(176,32)

Series: Derived Chief Lower central Upper central

C1C22 — D42D11
C1C11C22D22C4×D11 — D42D11
C11C22 — D42D11
C1C2D4

Generators and relations for D42D11
 G = < a,b,c,d | a4=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

2C2
2C2
22C2
11C4
11C22
11C4
11C4
2C22
2C22
2D11
11C2×C4
11D4
11D4
11C2×C4
11Q8
11C2×C4
11C4○D4

Smallest permutation representation of D42D11
On 88 points
Generators in S88
(1 65 21 54)(2 66 22 55)(3 56 12 45)(4 57 13 46)(5 58 14 47)(6 59 15 48)(7 60 16 49)(8 61 17 50)(9 62 18 51)(10 63 19 52)(11 64 20 53)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)
(1 76)(2 77)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)
(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 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(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)| (1,65,21,54)(2,66,22,55)(3,56,12,45)(4,57,13,46)(5,58,14,47)(6,59,15,48)(7,60,16,49)(8,61,17,50)(9,62,18,51)(10,63,19,52)(11,64,20,53)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77), (1,76)(2,77)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)>;

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

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

D42D11 is a maximal subgroup of
D4⋊D22  D83D11  D4.D22  Q8.D22  D46D22  C4○D4×D11  D4.10D22
D42D11 is a maximal quotient of
C23.11D22  C22⋊Dic22  C23.D22  Dic114D4  D22.D4  Dic11.D4  C22.D44  Dic22⋊C4  Dic11.Q8  C44.3Q8  C4⋊C47D11  D222Q8  C4⋊C4⋊D11  D4×Dic11  C23.18D22  C44.17D4  C442D4  Dic11⋊D4

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E11A···11E22A···22E22F···22O44A···44E
order122224444411···1122···2222···2244···44
size1122222111122222···22···24···44···4

35 irreducible representations

dim11111122224
type+++++++++-
imageC1C2C2C2C2C2C4○D4D11D22D22D42D11
kernelD42D11Dic22C4×D11C2×Dic11C11⋊D4D4×C11C11D4C4C22C1
# reps111221255105

Matrix representation of D42D11 in GL4(𝔽89) generated by

1000
0100
00340
002255
,
88000
08800
003481
002255
,
84100
647600
0010
0001
,
27800
658700
0010
005388
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,34,22,0,0,0,55],[88,0,0,0,0,88,0,0,0,0,34,22,0,0,81,55],[84,64,0,0,1,76,0,0,0,0,1,0,0,0,0,1],[2,65,0,0,78,87,0,0,0,0,1,53,0,0,0,88] >;

D42D11 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_{11}
% in TeX

G:=Group("D4:2D11");
// GroupNames label

G:=SmallGroup(176,32);
// by ID

G=gap.SmallGroup(176,32);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,46,182,97,4004]);
// 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,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D42D11 in TeX

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