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## G = Dic28order 112 = 24·7

### Dicyclic group

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

 Derived series C1 — C28 — Dic28
 Chief series C1 — C7 — C14 — C28 — Dic14 — Dic28
 Lower central C7 — C14 — C28 — Dic28
 Upper central C1 — C2 — C4 — C8

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

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

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

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

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

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 4A 4B 4C 7A 7B 7C 8A 8B 14A 14B 14C 28A ··· 28F 56A ··· 56L order 1 2 4 4 4 7 7 7 8 8 14 14 14 28 ··· 28 56 ··· 56 size 1 1 2 28 28 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

31 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 D4 D7 Q16 D14 D28 Dic28 kernel Dic28 C56 Dic14 C14 C8 C7 C4 C2 C1 # reps 1 1 2 1 3 2 3 6 12

Matrix representation of Dic28 in GL2(𝔽113) generated by

 40 106 69 84
,
 55 37 19 58
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

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