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G = D7×C28order 392 = 23·72

Direct product of C28 and D7

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C7 — D7×C28
C1C7C14C7×C14D7×C14 — D7×C28
C7 — D7×C28
C1C28

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

7C2
7C2
2C7
2C7
2C7
7C22
7C4
2C14
2C14
2C14
7C14
7C14
7C2×C4
2C28
2C28
2C28
7C2×C14
7C28
7C2×C28

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

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

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

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

140 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F7G···7AA14A···14F14G···14AA14AB···14AM28A···28L28M···28BB28BC···28BN
order122244447···77···714···1414···1414···1428···2828···2828···28
size117711771···12···21···12···27···71···12···27···7

140 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C7C14C14C14C28D7D14C4×D7C7×D7D7×C14D7×C28
kernelD7×C28C7×Dic7C7×C28D7×C14C7×D7C4×D7Dic7C28D14D7C28C14C7C4C2C1
# reps11114666624336181836

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

150
015
,
1914
1128
,
10
1128
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

Export

Subgroup lattice of D7×C28 in TeX

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