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G = C7×D28order 392 = 23·72

Direct product of C7 and D28

direct product, metacyclic, supersoluble, monomial

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

C1C14 — C7×D28
C1C7C14C7×C14D7×C14 — C7×D28
C7C14 — C7×D28
C1C14C28

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

14C2
14C2
2C7
2C7
2C7
7C22
7C22
2C14
2D7
2C14
2D7
2C14
14C14
14C14
7D4
2C28
2C28
2C28
7C2×C14
7C2×C14
2C7×D7
2C7×D7
7C7×D4

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

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

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

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

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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