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G = D445C2order 176 = 24·11

The semidirect product of D44 and C2 acting through Inn(D44)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D445C2, C4.16D22, Dic225C2, C22.4C23, C22.2D22, C44.16C22, D22.1C22, Dic11.2C22, (C2×C44)⋊4C2, (C2×C4)⋊3D11, (C4×D11)⋊4C2, C111(C4○D4), C11⋊D43C2, C2.5(C22×D11), (C2×C22).11C22, SmallGroup(176,30)

Series: Derived Chief Lower central Upper central

C1C22 — D445C2
C1C11C22D22C4×D11 — D445C2
C11C22 — D445C2
C1C4C2×C4

Generators and relations for D445C2
 G = < a,b,c | a44=b2=c2=1, bab=a-1, ac=ca, cbc=a22b >

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

Smallest permutation representation of D445C2
On 88 points
Generators in S88
(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 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 47)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 45)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)

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

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

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

D445C2 is a maximal subgroup of
D441C4  D444C4  D44.2C4  D887C2  D44.C4  C8⋊D22  C8.D22  D446C22  C44.C23  D4.8D22  D46D22  Q8.10D22  C4○D4×D11  D48D22  D4.10D22
D445C2 is a maximal quotient of
C4×Dic22  C44.6Q8  C42⋊D11  C4×D44  C4.D44  C422D11  C23.D22  D22.D4  D22⋊D4  Dic11.D4  Dic11.Q8  D22.5D4  D22⋊Q8  C4⋊C4⋊D11  C44.48D4  C23.21D22  C4×C11⋊D4  C23.23D22  C447D4

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E11A···11E22A···22O44A···44T
order122224444411···1122···2244···44
size112222211222222···22···22···2

50 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2C4○D4D11D22D22D445C2
kernelD445C2Dic22C4×D11D44C11⋊D4C2×C44C11C2×C4C4C22C1
# reps1121212510520

Matrix representation of D445C2 in GL2(𝔽89) generated by

5254
6088
,
375
2952
,
8027
309
G:=sub<GL(2,GF(89))| [52,60,54,88],[37,29,5,52],[80,30,27,9] >;

D445C2 in GAP, Magma, Sage, TeX

D_{44}\rtimes_5C_2
% in TeX

G:=Group("D44:5C2");
// GroupNames label

G:=SmallGroup(176,30);
// by ID

G=gap.SmallGroup(176,30);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,46,182,4004]);
// Polycyclic

G:=Group<a,b,c|a^44=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^22*b>;
// generators/relations

Export

Subgroup lattice of D445C2 in TeX

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