Copied to
clipboard

G = C7xC7:D4order 392 = 23·72

Direct product of C7 and C7:D4

direct product, metabelian, supersoluble, monomial

Aliases: C7xC7:D4, C14wrC2, C72:6D4, Dic7:C14, D14:2C14, C142:2C2, C14.21D14, C7:2(C7xD4), C22:(C7xD7), (C2xC14):1D7, (C2xC14):2C14, (D7xC14):4C2, C2.5(D7xC14), C14.5(C2xC14), (C7xDic7):4C2, (C7xC14).10C22, SmallGroup(392,27)

Series: Derived Chief Lower central Upper central

C1C14 — C7xC7:D4
C1C7C14C7xC14D7xC14 — C7xC7:D4
C7C14 — C7xC7:D4
C1C14C2xC14

Generators and relations for C7xC7: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 >

Subgroups: 130 in 47 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C22, C7, D4, D7, C14, D14, C2xC14, C7:D4, C7xD4, C7xD7, D7xC14, C7xC7:D4
2C2
14C2
2C7
2C7
2C7
7C22
7C4
2C14
2C14
2C14
2C14
2C14
2C14
2D7
2C14
2C14
2C14
2C14
2C14
14C14
7D4
2C2xC14
2C2xC14
2C2xC14
7C2xC14
7C28
2C7xD7
2C7xC14
7C7xD4

Permutation representations of C7xC7: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:D4C7xD4C7xD7D7xC14C7xC7:D4
kernelC7xC7:D4C7xDic7D7xC14C142C7:D4Dic7D14C2xC14C72C2xC14C14C7C7C22C2C1
# reps1111666613366181836

Matrix representation of C7xC7: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] >;

C7xC7: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 C7xC7:D4 in TeX

׿
x
:
Z
F
o
wr
Q
<