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G = C7xD8order 112 = 24·7

Direct product of C7 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C7xD8, D4:C14, C8:1C14, C56:5C2, C14.14D4, C28.17C22, (C7xD4):4C2, C2.3(C7xD4), C4.1(C2xC14), SmallGroup(112,24)

Series: Derived Chief Lower central Upper central

C1C4 — C7xD8
C1C2C4C28C7xD4 — C7xD8
C1C2C4 — C7xD8
C1C14C28 — C7xD8

Generators and relations for C7xD8
 G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 38 in 22 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C22, C7, D4, C14, D8, C2xC14, C7xD4, C7xD8
4C2
4C2
2C22
2C22
4C14
4C14
2C2xC14
2C2xC14

Smallest permutation representation of C7xD8
On 56 points
Generators in S56
(1 31 34 55 18 47 14)(2 32 35 56 19 48 15)(3 25 36 49 20 41 16)(4 26 37 50 21 42 9)(5 27 38 51 22 43 10)(6 28 39 52 23 44 11)(7 29 40 53 24 45 12)(8 30 33 54 17 46 13)
(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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 52)(50 51)(53 56)(54 55)

G:=sub<Sym(56)| (1,31,34,55,18,47,14)(2,32,35,56,19,48,15)(3,25,36,49,20,41,16)(4,26,37,50,21,42,9)(5,27,38,51,22,43,10)(6,28,39,52,23,44,11)(7,29,40,53,24,45,12)(8,30,33,54,17,46,13), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)>;

G:=Group( (1,31,34,55,18,47,14)(2,32,35,56,19,48,15)(3,25,36,49,20,41,16)(4,26,37,50,21,42,9)(5,27,38,51,22,43,10)(6,28,39,52,23,44,11)(7,29,40,53,24,45,12)(8,30,33,54,17,46,13), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,23)(21,22)(25,28)(26,27)(29,32)(30,31)(33,34)(35,40)(36,39)(37,38)(41,44)(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55) );

G=PermutationGroup([[(1,31,34,55,18,47,14),(2,32,35,56,19,48,15),(3,25,36,49,20,41,16),(4,26,37,50,21,42,9),(5,27,38,51,22,43,10),(6,28,39,52,23,44,11),(7,29,40,53,24,45,12),(8,30,33,54,17,46,13)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,18),(19,24),(20,23),(21,22),(25,28),(26,27),(29,32),(30,31),(33,34),(35,40),(36,39),(37,38),(41,44),(42,43),(45,48),(46,47),(49,52),(50,51),(53,56),(54,55)]])

C7xD8 is a maximal subgroup of   C7:D16  D8.D7  D8:D7  D8:3D7

49 conjugacy classes

class 1 2A2B2C 4 7A···7F8A8B14A···14F14G···14R28A···28F56A···56L
order122247···78814···1414···1428···2856···56
size114421···1221···14···42···22···2

49 irreducible representations

dim1111112222
type+++++
imageC1C2C2C7C14C14D4D8C7xD4C7xD8
kernelC7xD8C56C7xD4D8C8D4C14C7C2C1
# reps112661212612

Matrix representation of C7xD8 in GL2(F113) generated by

1060
0106
,
3182
3131
,
3182
8282
G:=sub<GL(2,GF(113))| [106,0,0,106],[31,31,82,31],[31,82,82,82] >;

C7xD8 in GAP, Magma, Sage, TeX

C_7\times D_8
% in TeX

G:=Group("C7xD8");
// GroupNames label

G:=SmallGroup(112,24);
// by ID

G=gap.SmallGroup(112,24);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,301,1683,848,58]);
// Polycyclic

G:=Group<a,b,c|a^7=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7xD8 in TeX

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