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## G = D7×Dic7order 392 = 23·72

### Direct product of D7 and Dic7

Aliases: D7×Dic7, D14.D7, C14.1D14, (C7×D7)⋊C4, C2.1D72, C73(C4×D7), (D7×C14).C2, C722(C2×C4), C7⋊Dic71C2, C71(C2×Dic7), (C7×Dic7)⋊2C2, (C7×C14).1C22, SmallGroup(392,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — D7×Dic7
 Chief series C1 — C7 — C72 — C7×C14 — D7×C14 — D7×Dic7
 Lower central C72 — D7×Dic7
 Upper central C1 — C2

Generators and relations for D7×Dic7
G = < a,b,c,d | a7=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D7×Dic7
On 56 points
Generators in S56
(1 3 5 7 9 11 13)(2 4 6 8 10 12 14)(15 27 25 23 21 19 17)(16 28 26 24 22 20 18)(29 41 39 37 35 33 31)(30 42 40 38 36 34 32)(43 45 47 49 51 53 55)(44 46 48 50 52 54 56)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 29)(13 30)(14 31)(15 53)(16 54)(17 55)(18 56)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)
(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 21 8 28)(2 20 9 27)(3 19 10 26)(4 18 11 25)(5 17 12 24)(6 16 13 23)(7 15 14 22)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 56 42 49)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 7G ··· 7O 14A ··· 14F 14G ··· 14O 14P ··· 14U 28A ··· 28F order 1 2 2 2 4 4 4 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 7 7 7 7 49 49 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 14 ··· 14 14 ··· 14

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 D7 D7 Dic7 D14 C4×D7 D72 D7×Dic7 kernel D7×Dic7 C7×Dic7 C7⋊Dic7 D7×C14 C7×D7 Dic7 D14 D7 C14 C7 C2 C1 # reps 1 1 1 1 4 3 3 6 6 6 9 9

Matrix representation of D7×Dic7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 10 5 0 0 9 22
,
 28 0 0 0 0 28 0 0 0 0 28 18 0 0 0 1
,
 1 22 0 0 7 10 0 0 0 0 1 0 0 0 0 1
,
 17 0 0 0 3 12 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,10,9,0,0,5,22],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,18,1],[1,7,0,0,22,10,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;

D7×Dic7 in GAP, Magma, Sage, TeX

D_7\times {\rm Dic}_7
% in TeX

G:=Group("D7xDic7");
// GroupNames label

G:=SmallGroup(392,18);
// by ID

G=gap.SmallGroup(392,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,26,488,8404]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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