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G = Dic28order 112 = 24·7

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic28, C8.D7, C71Q16, C56.1C2, C14.3D4, C2.5D28, C4.10D14, C28.10C22, Dic14.1C2, SmallGroup(112,7)

Series: Derived Chief Lower central Upper central

C1C28 — Dic28
C1C7C14C28Dic14 — Dic28
C7C14C28 — Dic28
C1C2C4C8

Generators and relations for Dic28
 G = < a,b | a56=1, b2=a28, bab-1=a-1 >

14C4
14C4
7Q8
7Q8
2Dic7
2Dic7
7Q16

Smallest permutation representation of Dic28
Regular action 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 62 29 90)(2 61 30 89)(3 60 31 88)(4 59 32 87)(5 58 33 86)(6 57 34 85)(7 112 35 84)(8 111 36 83)(9 110 37 82)(10 109 38 81)(11 108 39 80)(12 107 40 79)(13 106 41 78)(14 105 42 77)(15 104 43 76)(16 103 44 75)(17 102 45 74)(18 101 46 73)(19 100 47 72)(20 99 48 71)(21 98 49 70)(22 97 50 69)(23 96 51 68)(24 95 52 67)(25 94 53 66)(26 93 54 65)(27 92 55 64)(28 91 56 63)

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

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

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

Dic28 is a maximal subgroup of
C112⋊C2  Dic56  D8.D7  C7⋊Q32  D567C2  C8.D14  D83D7  SD16⋊D7  D7×Q16  C8.F7  C3⋊Dic28  Dic84
Dic28 is a maximal quotient of
C28.44D4  C561C4  C3⋊Dic28  Dic84

31 conjugacy classes

class 1  2 4A4B4C7A7B7C8A8B14A14B14C28A···28F56A···56L
order124447778814141428···2856···56
size1122828222222222···22···2

31 irreducible representations

dim111222222
type+++++-++-
imageC1C2C2D4D7Q16D14D28Dic28
kernelDic28C56Dic14C14C8C7C4C2C1
# reps1121323612

Matrix representation of Dic28 in GL2(𝔽113) generated by

40106
6984
,
5537
1958
G:=sub<GL(2,GF(113))| [40,69,106,84],[55,19,37,58] >;

Dic28 in GAP, Magma, Sage, TeX

{\rm Dic}_{28}
% in TeX

G:=Group("Dic28");
// GroupNames label

G:=SmallGroup(112,7);
// by ID

G=gap.SmallGroup(112,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,66,182,42,2404]);
// Polycyclic

G:=Group<a,b|a^56=1,b^2=a^28,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic28 in TeX

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