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G = C7×D28  order 392 = 23·72

Direct product of C7 and D28

direct product, metacyclic, supersoluble, monomial

Aliases: C7×D28, C28⋊3D7, C28⋊1C14, C72⋊4D4, D14⋊1C14, C14.19D14, C4⋊(C7×D7), C7⋊1(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 >

C1
C2
14C2
14C2
C7
C7
2C7
2C7
2C7
C4
7C22
7C22
C14
C14
2C14
2D7
2C14
2D7
2C14
14C14
14C14
C72
7D4
C28
C28
D14
D14
2C28
2C28
2C28
7C2×C14
7C2×C14
C7×C14
2C7×D7
2C7×D7
D28
7C7×D4
D7×C14
C7×C28
D7×C14
C7×D28

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 2A2B2C 4 7A···7F7G···7AA14A···14F14G···14AA14AB···14AM28A···28AV
order122247···77···714···1414···1414···1428···28
size11141421···12···21···12···214···142···2

119 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C7C14C14D4D7D14D28C7×D4C7×D7D7×C14C7×D28
kernelC7×D28C7×C28D7×C14D28C28D14C72C28C14C7C7C4C2C1
# reps112661213366181836

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

240
024
,
210
018
,
018
210
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
 

Export

Subgroup lattice of C7×D28 in TeX

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