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## G = C7×D28order 392 = 23·72

### Direct product of C7 and D28

Aliases: C7×D28, C283D7, C281C14, C724D4, D141C14, C14.19D14, C4⋊(C7×D7), C71(C7×D4), (C7×C28)⋊2C2, (D7×C14)⋊3C2, C2.4(D7×C14), C14.3(C2×C14), (C7×C14).8C22, SmallGroup(392,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C7×D28
 Chief series C1 — C7 — C14 — C7×C14 — D7×C14 — C7×D28
 Lower central C7 — C14 — C7×D28
 Upper central C1 — C14 — C28

Generators and relations for C7×D28
G = < a,b,c | a7=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C7×D28
On 56 points
Generators in S56
(1 13 25 9 21 5 17)(2 14 26 10 22 6 18)(3 15 27 11 23 7 19)(4 16 28 12 24 8 20)(29 45 33 49 37 53 41)(30 46 34 50 38 54 42)(31 47 35 51 39 55 43)(32 48 36 52 40 56 44)
(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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 56)(25 55)(26 54)(27 53)(28 52)

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

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

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

119 conjugacy classes

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

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 D4 D7 D14 D28 C7×D4 C7×D7 D7×C14 C7×D28 kernel C7×D28 C7×C28 D7×C14 D28 C28 D14 C72 C28 C14 C7 C7 C4 C2 C1 # reps 1 1 2 6 6 12 1 3 3 6 6 18 18 36

Matrix representation of C7×D28 in GL2(𝔽29) generated by

 24 0 0 24
,
 21 0 0 18
,
 0 18 21 0
G:=sub<GL(2,GF(29))| [24,0,0,24],[21,0,0,18],[0,21,18,0] >;

C7×D28 in GAP, Magma, Sage, TeX

C_7\times D_{28}
% in TeX

G:=Group("C7xD28");
// GroupNames label

G:=SmallGroup(392,25);
// by ID

G=gap.SmallGroup(392,25);
# by ID

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

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

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