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G = C7×Q8order 56 = 23·7

Direct product of C7 and Q8

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

Aliases: C7×Q8, C4.C14, C28.3C2, C14.7C22, C2.2(C2×C14), SmallGroup(56,10)

Series: Derived Chief Lower central Upper central

C1C2 — C7×Q8
C1C2C14C28 — C7×Q8
C1C2 — C7×Q8
C1C14 — C7×Q8

Generators and relations for C7×Q8
 G = < a,b,c | a7=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of C7×Q8
Regular action on 56 points
Generators in S56
(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 29 15 24)(2 30 16 25)(3 31 17 26)(4 32 18 27)(5 33 19 28)(6 34 20 22)(7 35 21 23)(8 41 55 46)(9 42 56 47)(10 36 50 48)(11 37 51 49)(12 38 52 43)(13 39 53 44)(14 40 54 45)
(1 43 15 38)(2 44 16 39)(3 45 17 40)(4 46 18 41)(5 47 19 42)(6 48 20 36)(7 49 21 37)(8 32 55 27)(9 33 56 28)(10 34 50 22)(11 35 51 23)(12 29 52 24)(13 30 53 25)(14 31 54 26)

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

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

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

C7×Q8 is a maximal subgroup of   Q8⋊D7  C7⋊Q16  Q82D7  C14.A4

35 conjugacy classes

class 1  2 4A4B4C7A···7F14A···14F28A···28R
order124447···714···1428···28
size112221···11···12···2

35 irreducible representations

dim111122
type++-
imageC1C2C7C14Q8C7×Q8
kernelC7×Q8C28Q8C4C7C1
# reps1361816

Matrix representation of C7×Q8 in GL2(𝔽29) generated by

160
016
,
01
280
,
119
918
G:=sub<GL(2,GF(29))| [16,0,0,16],[0,28,1,0],[11,9,9,18] >;

C7×Q8 in GAP, Magma, Sage, TeX

C_7\times Q_8
% in TeX

G:=Group("C7xQ8");
// GroupNames label

G:=SmallGroup(56,10);
// by ID

G=gap.SmallGroup(56,10);
# by ID

G:=PCGroup([4,-2,-2,-7,-2,112,241,117]);
// Polycyclic

G:=Group<a,b,c|a^7=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Q8 in TeX

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