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G = C112order 112 = 24·7

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C112, also denoted Z112, SmallGroup(112,2)

Series: Derived Chief Lower central Upper central

C1 — C112
C1C2C4C8C56 — C112
C1 — C112
C1 — C112

Generators and relations for C112
 G = < a | a112=1 >


Smallest permutation representation of C112
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)

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)>;

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) );

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)])

C112 is a maximal subgroup of   C7⋊C32  C16⋊D7  D112  C112⋊C2  Dic56

112 conjugacy classes

class 1  2 4A4B7A···7F8A8B8C8D14A···14F16A···16H28A···28L56A···56X112A···112AV
order12447···7888814···1416···1628···2856···56112···112
size11111···111111···11···11···11···11···1

112 irreducible representations

dim1111111111
type++
imageC1C2C4C7C8C14C16C28C56C112
kernelC112C56C28C16C14C8C7C4C2C1
# reps1126468122448

Matrix representation of C112 in GL2(𝔽41) generated by

1423
3925
G:=sub<GL(2,GF(41))| [14,39,23,25] >;

C112 in GAP, Magma, Sage, TeX

C_{112}
% in TeX

G:=Group("C112");
// GroupNames label

G:=SmallGroup(112,2);
// by ID

G=gap.SmallGroup(112,2);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-2,70,42,58]);
// Polycyclic

G:=Group<a|a^112=1>;
// generators/relations

Export

Subgroup lattice of C112 in TeX

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