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## G = Dic7⋊2D7order 392 = 23·72

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

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

 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 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

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

50 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C4 D7 D14 C4×D7 D72 Dic7⋊2D7 kernel Dic7⋊2D7 C7×Dic7 C2×C7⋊D7 C7⋊D7 Dic7 C14 C7 C2 C1 # reps 1 2 1 4 6 6 12 9 9

Matrix representation of Dic72D7 in GL4(𝔽29) generated by

 28 1 0 0 20 8 0 0 0 0 28 0 0 0 0 28
,
 21 1 0 0 24 8 0 0 0 0 17 0 0 0 0 17
,
 1 0 0 0 0 1 0 0 0 0 21 7 0 0 24 26
,
 8 28 0 0 5 21 0 0 0 0 1 28 0 0 0 28
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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