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G = Dic72D7order 392 = 23·72

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

metabelian, supersoluble, monomial, A-group

Aliases: Dic72D7, C14.2D14, C2.2D72, C7⋊D71C4, C71(C4×D7), C723(C2×C4), (C7×Dic7)⋊3C2, (C7×C14).2C22, (C2×C7⋊D7).1C2, SmallGroup(392,19)

Series: Derived Chief Lower central Upper central

C1C72 — Dic72D7
C1C7C72C7×C14C7×Dic7 — Dic72D7
C72 — Dic72D7
C1C2

Generators and relations for Dic72D7
 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 >

49C2
49C2
2C7
2C7
2C7
7C4
7C4
49C22
2C14
2C14
2C14
7D7
7D7
7D7
7D7
14D7
14D7
14D7
14D7
14D7
14D7
49C2×C4
7C28
7D14
7C28
7D14
14D14
14D14
14D14
7C4×D7
7C4×D7

Permutation representations of Dic72D7
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 19 8 26)(2 18 9 25)(3 17 10 24)(4 16 11 23)(5 15 12 22)(6 28 13 21)(7 27 14 20)
(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 25)(16 24)(17 23)(18 22)(19 21)(26 28)

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

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

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

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×D7D72Dic72D7
kernelDic72D7C7×Dic7C2×C7⋊D7C7⋊D7Dic7C14C7C2C1
# reps1214661299

Matrix representation of Dic72D7 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] >;

Dic72D7 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

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Subgroup lattice of Dic72D7 in TeX

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