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G = D4.D11order 176 = 24·11

The non-split extension by D4 of D11 acting via D11/C11=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D11, C22.8D4, C4.2D22, C112SD16, Dic222C2, C44.2C22, C11⋊C82C2, (D4×C11).1C2, C2.5(C11⋊D4), SmallGroup(176,15)

Series: Derived Chief Lower central Upper central

C1C44 — D4.D11
C1C11C22C44Dic22 — D4.D11
C11C22C44 — D4.D11
C1C2C4D4

Generators and relations for D4.D11
 G = < a,b,c,d | a4=b2=c11=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
22C4
4C22
11C8
11Q8
2Dic11
2C2×C22
11SD16

Smallest permutation representation of D4.D11
On 88 points
Generators in S88
(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 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 66 21 55)(2 65 22 54)(3 64 12 53)(4 63 13 52)(5 62 14 51)(6 61 15 50)(7 60 16 49)(8 59 17 48)(9 58 18 47)(10 57 19 46)(11 56 20 45)(23 86 34 75)(24 85 35 74)(25 84 36 73)(26 83 37 72)(27 82 38 71)(28 81 39 70)(29 80 40 69)(30 79 41 68)(31 78 42 67)(32 88 43 77)(33 87 44 76)

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

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

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

D4.D11 is a maximal subgroup of   D4⋊D22  D83D11  SD16×D11  D4.D22  D446C22  D4.8D22  D4.9D22
D4.D11 is a maximal quotient of   C4.Dic22  C22.Q16  D4⋊Dic11

32 conjugacy classes

class 1 2A2B4A4B8A8B11A···11E22A···22E22F···22O44A···44E
order122448811···1122···2222···2244···44
size11424422222···22···24···44···4

32 irreducible representations

dim1111222224
type+++++++-
imageC1C2C2C2D4SD16D11D22C11⋊D4D4.D11
kernelD4.D11C11⋊C8Dic22D4×C11C22C11D4C4C2C1
# reps11111255105

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

1000
0100
008885
00451
,
1000
0100
0010
004488
,
0100
888600
0010
0001
,
683400
82100
00080
00100
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,88,45,0,0,85,1],[1,0,0,0,0,1,0,0,0,0,1,44,0,0,0,88],[0,88,0,0,1,86,0,0,0,0,1,0,0,0,0,1],[68,8,0,0,34,21,0,0,0,0,0,10,0,0,80,0] >;

D4.D11 in GAP, Magma, Sage, TeX

D_4.D_{11}
% in TeX

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

G:=SmallGroup(176,15);
// by ID

G=gap.SmallGroup(176,15);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,40,61,182,97,42,4004]);
// Polycyclic

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

Export

Subgroup lattice of D4.D11 in TeX

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