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## G = C7×Dic14order 392 = 23·72

### Direct product of C7 and Dic14

Aliases: C7×Dic14, C723Q8, C28.7D7, C28.1C14, Dic7.C14, C14.17D14, C7⋊(C7×Q8), C4.(C7×D7), (C7×C28).2C2, C2.3(D7×C14), C14.1(C2×C14), (C7×C14).6C22, (C7×Dic7).2C2, SmallGroup(392,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C7×Dic14
 Chief series C1 — C7 — C14 — C7×C14 — C7×Dic7 — C7×Dic14
 Lower central C7 — C14 — C7×Dic14
 Upper central C1 — C14 — C28

Generators and relations for C7×Dic14
G = < a,b,c | a7=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic14
On 56 points
Generators in S56
(1 5 9 13 17 21 25)(2 6 10 14 18 22 26)(3 7 11 15 19 23 27)(4 8 12 16 20 24 28)(29 53 49 45 41 37 33)(30 54 50 46 42 38 34)(31 55 51 47 43 39 35)(32 56 52 48 44 40 36)
(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 36 15 50)(2 35 16 49)(3 34 17 48)(4 33 18 47)(5 32 19 46)(6 31 20 45)(7 30 21 44)(8 29 22 43)(9 56 23 42)(10 55 24 41)(11 54 25 40)(12 53 26 39)(13 52 27 38)(14 51 28 37)

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

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

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

119 conjugacy classes

 class 1 2 4A 4B 4C 7A ··· 7F 7G ··· 7AA 14A ··· 14F 14G ··· 14AA 28A ··· 28AV 28AW ··· 28BH order 1 2 4 4 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 14 14 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 14 ··· 14

119 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C7 C14 C14 Q8 D7 D14 Dic14 C7×Q8 C7×D7 D7×C14 C7×Dic14 kernel C7×Dic14 C7×Dic7 C7×C28 Dic14 Dic7 C28 C72 C28 C14 C7 C7 C4 C2 C1 # reps 1 2 1 6 12 6 1 3 3 6 6 18 18 36

Matrix representation of C7×Dic14 in GL2(𝔽29) generated by

 7 0 0 7
,
 21 0 0 18
,
 0 1 28 0
G:=sub<GL(2,GF(29))| [7,0,0,7],[21,0,0,18],[0,28,1,0] >;

C7×Dic14 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_{14}
% in TeX

G:=Group("C7xDic14");
// GroupNames label

G:=SmallGroup(392,23);
// by ID

G=gap.SmallGroup(392,23);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,140,301,146,8404]);
// Polycyclic

G:=Group<a,b,c|a^7=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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