Copied to
clipboard

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

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

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

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

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

׿
×
𝔽