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G = D112order 224 = 25·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D112, C71D16, C161D7, C1121C2, D561C2, C4.1D28, C2.3D56, C14.1D8, C28.24D4, C8.13D14, C56.14C22, sometimes denoted D224 or Dih112 or Dih224, SmallGroup(224,5)

Series: Derived Chief Lower central Upper central

C1C56 — D112
C1C7C14C28C56D56 — D112
C7C14C28C56 — D112
C1C2C4C8C16

Generators and relations for D112
 G = < a,b | a112=b2=1, bab=a-1 >

56C2
56C2
28C22
28C22
8D7
8D7
14D4
14D4
4D14
4D14
7D8
7D8
2D28
2D28
7D16

Smallest permutation representation of D112
On 112 points
Generators in S112
(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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)

G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,100),(14,99),(15,98),(16,97),(17,96),(18,95),(19,94),(20,93),(21,92),(22,91),(23,90),(24,89),(25,88),(26,87),(27,86),(28,85),(29,84),(30,83),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57)])

D112 is a maximal subgroup of
D224  C224⋊C2  C7⋊D32  C7⋊SD64  D1127C2  C16⋊D14  D7×D16  D112⋊C2  Q323D7
D112 is a maximal quotient of
D224  C224⋊C2  Dic112  C1125C4  C2.D112

59 conjugacy classes

class 1 2A2B2C 4 7A7B7C8A8B14A14B14C16A16B16C16D28A···28F56A···56L112A···112X
order12224777881414141616161628···2856···56112···112
size11565622222222222222···22···22···2

59 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2D4D7D8D14D16D28D56D112
kernelD112C112D56C28C16C14C8C7C4C2C1
# reps1121323461224

Matrix representation of D112 in GL2(𝔽113) generated by

7435
7814
,
7435
539
G:=sub<GL(2,GF(113))| [74,78,35,14],[74,5,35,39] >;

D112 in GAP, Magma, Sage, TeX

D_{112}
% in TeX

G:=Group("D112");
// GroupNames label

G:=SmallGroup(224,5);
// by ID

G=gap.SmallGroup(224,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,79,218,122,579,69,6917]);
// Polycyclic

G:=Group<a,b|a^112=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D112 in TeX

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