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G = C7×D8order 112 = 24·7

Direct product of C7 and D8

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

Aliases: C7×D8, D4⋊C14, C81C14, C565C2, C14.14D4, C28.17C22, (C7×D4)⋊4C2, C2.3(C7×D4), C4.1(C2×C14), SmallGroup(112,24)

Series: Derived Chief Lower central Upper central

C1C4 — C7×D8
C1C2C4C28C7×D4 — C7×D8
C1C2C4 — C7×D8
C1C14C28 — C7×D8

Generators and relations for C7×D8
 G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C14
4C14
2C2×C14
2C2×C14

Smallest permutation representation of C7×D8
On 56 points
Generators in S56
(1 31 34 55 22 47 10)(2 32 35 56 23 48 11)(3 25 36 49 24 41 12)(4 26 37 50 17 42 13)(5 27 38 51 18 43 14)(6 28 39 52 19 44 15)(7 29 40 53 20 45 16)(8 30 33 54 21 46 9)
(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)| (1,31,34,55,22,47,10)(2,32,35,56,23,48,11)(3,25,36,49,24,41,12)(4,26,37,50,17,42,13)(5,27,38,51,18,43,14)(6,28,39,52,19,44,15)(7,29,40,53,20,45,16)(8,30,33,54,21,46,9), (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( (1,31,34,55,22,47,10)(2,32,35,56,23,48,11)(3,25,36,49,24,41,12)(4,26,37,50,17,42,13)(5,27,38,51,18,43,14)(6,28,39,52,19,44,15)(7,29,40,53,20,45,16)(8,30,33,54,21,46,9), (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([(1,31,34,55,22,47,10),(2,32,35,56,23,48,11),(3,25,36,49,24,41,12),(4,26,37,50,17,42,13),(5,27,38,51,18,43,14),(6,28,39,52,19,44,15),(7,29,40,53,20,45,16),(8,30,33,54,21,46,9)], [(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)])

C7×D8 is a maximal subgroup of   C7⋊D16  D8.D7  D8⋊D7  D83D7

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+++++
imageC1C2C2C7C14C14D4D8C7×D4C7×D8
kernelC7×D8C56C7×D4D8C8D4C14C7C2C1
# reps112661212612

Matrix representation of C7×D8 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] >;

C7×D8 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 C7×D8 in TeX

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