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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

 Derived series C1 — C2 — C7×Q8
 Chief series C1 — C2 — C14 — C28 — C7×Q8
 Lower central C1 — C2 — C7×Q8
 Upper central C1 — C14 — 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 33 19 24)(2 34 20 25)(3 35 21 26)(4 29 15 27)(5 30 16 28)(6 31 17 22)(7 32 18 23)(8 41 55 43)(9 42 56 44)(10 36 50 45)(11 37 51 46)(12 38 52 47)(13 39 53 48)(14 40 54 49)
(1 47 19 38)(2 48 20 39)(3 49 21 40)(4 43 15 41)(5 44 16 42)(6 45 17 36)(7 46 18 37)(8 29 55 27)(9 30 56 28)(10 31 50 22)(11 32 51 23)(12 33 52 24)(13 34 53 25)(14 35 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), (1,33,19,24)(2,34,20,25)(3,35,21,26)(4,29,15,27)(5,30,16,28)(6,31,17,22)(7,32,18,23)(8,41,55,43)(9,42,56,44)(10,36,50,45)(11,37,51,46)(12,38,52,47)(13,39,53,48)(14,40,54,49), (1,47,19,38)(2,48,20,39)(3,49,21,40)(4,43,15,41)(5,44,16,42)(6,45,17,36)(7,46,18,37)(8,29,55,27)(9,30,56,28)(10,31,50,22)(11,32,51,23)(12,33,52,24)(13,34,53,25)(14,35,54,26)>;

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

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

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

35 conjugacy classes

 class 1 2 4A 4B 4C 7A ··· 7F 14A ··· 14F 28A ··· 28R order 1 2 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2

35 irreducible representations

 dim 1 1 1 1 2 2 type + + - image C1 C2 C7 C14 Q8 C7×Q8 kernel C7×Q8 C28 Q8 C4 C7 C1 # reps 1 3 6 18 1 6

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

 16 0 0 16
,
 0 1 28 0
,
 11 9 9 18
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

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