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G = C7×C7⋊D4  order 392 = 23·72

Direct product of C7 and C7⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C7×C7⋊D4, C14≀C2, C72⋊6D4, Dic7⋊C14, D14⋊2C14, C142⋊2C2, C14.21D14, C7⋊2(C7×D4), C22⋊(C7×D7), (C2×C14)⋊1D7, (C2×C14)⋊2C14, (D7×C14)⋊4C2, C2.5(D7×C14), C14.5(C2×C14), (C7×Dic7)⋊4C2, (C7×C14).10C22, SmallGroup(392,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C7×C7⋊D4
Chief series
C1 — C7 — C14 — C7×C14 — D7×C14 — C7×C7⋊D4
Lower central
C7 — C14 — C7×C7⋊D4
Upper central
C1 — C14 — C2×C14

Generators and relations for C7×C7⋊D4
 G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

C1
C2
2C2
14C2
C7
C7
2C7
2C7
2C7
C22
7C22
7C4
C14
C14
2C14
2C14
2C14
2C14
2C14
2C14
2D7
2C14
2C14
2C14
2C14
2C14
14C14
C72
7D4
C2×C14
C2×C14
Dic7
D14
2C2×C14
2C2×C14
2C2×C14
7C2×C14
7C28
C7×C14
2C7×D7
2C7×C14
C7⋊D4
7C7×D4
D7×C14
C7×Dic7
C142
C7×C7⋊D4

Permutation representations of C7×C7⋊D4
►On 28 points - transitive group 28T47
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 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)

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

(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)>;
 
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), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28) );
 
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)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,21,20,19,18,17,16),(22,28,27,26,25,24,23)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)]])
 
G:=TransitiveGroup(28,47);
 

119 conjugacy classes

class 1 2A2B2C 4 7A···7F7G···7AA14A···14F14G···14BW14BX···14CC28A···28F
order122247···77···714···1414···1414···1428···28
size11214141···12···21···12···214···1414···14

119 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C7C14C14C14D4D7D14C7⋊D4C7×D4C7×D7D7×C14C7×C7⋊D4
kernelC7×C7⋊D4C7×Dic7D7×C14C142C7⋊D4Dic7D14C2×C14C72C2×C14C14C7C7C22C2C1
# reps1111666613366181836

Matrix representation of C7×C7⋊D4 ►in GL2(𝔽29) generated by

200
020
,
200
016
,
01
280
,
01
10
G:=sub<GL(2,GF(29))| [20,0,0,20],[20,0,0,16],[0,28,1,0],[0,1,1,0] >;
 

C7×C7⋊D4 in GAP, Magma, Sage, TeX 

C_7\times C_7\rtimes D_4
 
% in TeX
 
G:=Group("C7xC7:D4");
 
// GroupNames label
 
G:=SmallGroup(392,27);
 
// by ID
 
G=gap.SmallGroup(392,27);
 
# by ID
 
G:=PCGroup([5,-2,-2,-7,-2,-7,301,8404]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C7×C7⋊D4 in TeX

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