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G = Dic7⋊2D7  order 392 = 23·72

The semidirect product of Dic7 and D7 acting through Inn(Dic7)

metabelian, supersoluble, monomial, A-group

Aliases: Dic7⋊2D7, C14.2D14, C2.2D72, C7⋊D7⋊1C4, C7⋊1(C4×D7), C72⋊3(C2×C4), (C7×Dic7)⋊3C2, (C7×C14).2C22, (C2×C7⋊D7).1C2, SmallGroup(392,19)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — Dic7⋊2D7
Chief series
C1 — C7 — C72 — C7×C14 — C7×Dic7 — Dic7⋊2D7
Lower central
C72 — Dic7⋊2D7
Upper central
C1 — C2

Generators and relations for Dic7⋊2D7
 G = < a,b,c,d | a14=c7=d2=1, b2=a7, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

C1
C2
49C2
49C2
C7
C7
2C7
2C7
2C7
7C4
7C4
49C22
C14
C14
2C14
2C14
2C14
7D7
7D7
7D7
7D7
14D7
14D7
14D7
14D7
14D7
14D7
C72
49C2×C4
Dic7
Dic7
7C28
7D14
7C28
7D14
14D14
14D14
14D14
C7⋊D7
C7⋊D7
C7×C14
7C4×D7
7C4×D7
C7×Dic7
C2×C7⋊D7
C7×Dic7
Dic7⋊2D7

Permutation representations of Dic7⋊2D7
►On 28 points - transitive group 28T51
Generators in S28
(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)

(1 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)

(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)

(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 17)(18 28)(19 27)(20 26)(21 25)(22 24)
 
G:=sub<Sym(28)| (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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,17)(18,28)(19,27)(20,26)(21,25)(22,24)>;
 
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), (1,22,8,15)(2,21,9,28)(3,20,10,27)(4,19,11,26)(5,18,12,25)(6,17,13,24)(7,16,14,23), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,17)(18,28)(19,27)(20,26)(21,25)(22,24) );
 
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)], [(1,22,8,15),(2,21,9,28),(3,20,10,27),(4,19,11,26),(5,18,12,25),(6,17,13,24),(7,16,14,23)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,17),(18,28),(19,27),(20,26),(21,25),(22,24)]])
 
G:=TransitiveGroup(28,51);
 

50 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F7G···7O14A···14F14G···14O28A···28L
order122244447···77···714···1414···1428···28
size11494977772···24···42···24···414···14

50 irreducible representations

dim111122244
type+++++++
imageC1C2C2C4D7D14C4×D7D72Dic7⋊2D7
kernelDic7⋊2D7C7×Dic7C2×C7⋊D7C7⋊D7Dic7C14C7C2C1
# reps1214661299

Matrix representation of Dic7⋊2D7 ►in GL4(𝔽29) generated by

28100
20800
00280
00028
,
21100
24800
00170
00017
,
1000
0100
00217
002426
,
82800
52100
00128
00028
G:=sub<GL(4,GF(29))| [28,20,0,0,1,8,0,0,0,0,28,0,0,0,0,28],[21,24,0,0,1,8,0,0,0,0,17,0,0,0,0,17],[1,0,0,0,0,1,0,0,0,0,21,24,0,0,7,26],[8,5,0,0,28,21,0,0,0,0,1,0,0,0,28,28] >;
 

Dic7⋊2D7 in GAP, Magma, Sage, TeX 

{\rm Dic}_7\rtimes_2D_7
 
% in TeX
 
G:=Group("Dic7:2D7");
 
// GroupNames label
 
G:=SmallGroup(392,19);
 
// by ID
 
G=gap.SmallGroup(392,19);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-7,-7,20,26,488,8404]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^14=c^7=d^2=1,b^2=a^7,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Dic7⋊2D7 in TeX

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