Copied to
clipboard

G = C7⋊D28  order 392 = 23·72

The semidirect product of C7 and D28 acting via D28/D14=C2

metabelian, supersoluble, monomial

Aliases: C7⋊2D28, Dic7⋊D7, C72⋊3D4, D14⋊2D7, C14.4D14, C2.4D72, (D7×C14)⋊2C2, C7⋊1(C7⋊D4), (C7×Dic7)⋊1C2, (C7×C14).4C22, (C2×C7⋊D7)⋊1C2, SmallGroup(392,21)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7×C14 — C7⋊D28
Chief series
C1 — C7 — C72 — C7×C14 — D7×C14 — C7⋊D28
Lower central
C72 — C7×C14 — C7⋊D28
Upper central
C1 — C2

Generators and relations for C7⋊D28
 G = < a,b,c | a7=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >

C1
C2
14C2
98C2
C7
C7
2C7
2C7
2C7
7C22
7C4
49C22
C14
C14
2D7
2C14
2C14
2C14
14D7
14D7
14C14
14D7
14D7
14D7
14D7
14D7
14D7
C72
49D4
D14
Dic7
7C28
7C2×C14
7D14
7D14
14D14
14D14
14D14
C7×C14
2C7⋊D7
2C7×D7
7C7⋊D4
7D28
D7×C14
C2×C7⋊D7
C7×Dic7
C7⋊D28

Permutation representations of C7⋊D28
►On 28 points - transitive group 28T50
Generators in S28
(1 13 25 9 21 5 17)(2 18 6 22 10 26 14)(3 15 27 11 23 7 19)(4 20 8 24 12 28 16)

(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)
 
G:=sub<Sym(28)| (1,13,25,9,21,5,17)(2,18,6,22,10,26,14)(3,15,27,11,23,7,19)(4,20,8,24,12,28,16), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)>;
 
G:=Group( (1,13,25,9,21,5,17)(2,18,6,22,10,26,14)(3,15,27,11,23,7,19)(4,20,8,24,12,28,16), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19) );
 
G=PermutationGroup([[(1,13,25,9,21,5,17),(2,18,6,22,10,26,14),(3,15,27,11,23,7,19),(4,20,8,24,12,28,16)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19)]])
 
G:=TransitiveGroup(28,50);
 

47 conjugacy classes

class 1 2A2B2C 4 7A···7F7G···7O14A···14F14G···14O14P···14U28A···28F
order122247···77···714···1414···1414···1428···28
size111498142···24···42···24···414···1414···14

47 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2D4D7D7D14D28C7⋊D4D72C7⋊D28
kernelC7⋊D28C7×Dic7D7×C14C2×C7⋊D7C72Dic7D14C14C7C7C2C1
# reps111113366699

Matrix representation of C7⋊D28 ►in GL6(𝔽29)

100000
010000
00211800
00192600
000010
000001
,
2110000
2010000
00261100
0023300
00002827
000011
,
0190000
2600000
00261100
0023300
0000280
000011

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,19,0,0,0,0,18,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,20,0,0,0,0,1,1,0,0,0,0,0,0,26,23,0,0,0,0,11,3,0,0,0,0,0,0,28,1,0,0,0,0,27,1],[0,26,0,0,0,0,19,0,0,0,0,0,0,0,26,23,0,0,0,0,11,3,0,0,0,0,0,0,28,1,0,0,0,0,0,1] >;
 

C7⋊D28 in GAP, Magma, Sage, TeX 

C_7\rtimes D_{28}
 
% in TeX
 
G:=Group("C7:D28");
 
// GroupNames label
 
G:=SmallGroup(392,21);
 
// by ID
 
G=gap.SmallGroup(392,21);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-7,-7,61,26,488,8404]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C7⋊D28 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁