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## G = D7×C28order 392 = 23·72

### Direct product of C28 and D7

Aliases: D7×C28, C282C14, D14.C14, Dic72C14, C14.18D14, (C7×C28)⋊3C2, C71(C2×C28), C724(C2×C4), C2.1(D7×C14), C14.2(C2×C14), (C7×Dic7)⋊5C2, (D7×C14).2C2, (C7×C14).7C22, SmallGroup(392,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C28
 Chief series C1 — C7 — C14 — C7×C14 — D7×C14 — D7×C28
 Lower central C7 — D7×C28
 Upper central C1 — C28

Generators and relations for D7×C28
G = < a,b,c | a28=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C28
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 9 17 25 5 13 21)(2 10 18 26 6 14 22)(3 11 19 27 7 15 23)(4 12 20 28 8 16 24)(29 49 41 33 53 45 37)(30 50 42 34 54 46 38)(31 51 43 35 55 47 39)(32 52 44 36 56 48 40)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 7G ··· 7AA 14A ··· 14F 14G ··· 14AA 14AB ··· 14AM 28A ··· 28L 28M ··· 28BB 28BC ··· 28BN order 1 2 2 2 4 4 4 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 size 1 1 7 7 1 1 7 7 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 7 ··· 7 1 ··· 1 2 ··· 2 7 ··· 7

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D7 D14 C4×D7 C7×D7 D7×C14 D7×C28 kernel D7×C28 C7×Dic7 C7×C28 D7×C14 C7×D7 C4×D7 Dic7 C28 D14 D7 C28 C14 C7 C4 C2 C1 # reps 1 1 1 1 4 6 6 6 6 24 3 3 6 18 18 36

Matrix representation of D7×C28 in GL2(𝔽29) generated by

 15 0 0 15
,
 19 14 11 28
,
 1 0 11 28
G:=sub<GL(2,GF(29))| [15,0,0,15],[19,11,14,28],[1,11,0,28] >;

D7×C28 in GAP, Magma, Sage, TeX

D_7\times C_{28}
% in TeX

G:=Group("D7xC28");
// GroupNames label

G:=SmallGroup(392,24);
// by ID

G=gap.SmallGroup(392,24);
# by ID

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

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

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