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G = C7×C7⋊C3order 147 = 3·72

Direct product of C7 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C7×C7⋊C3, C7⋊C21, C721C3, SmallGroup(147,3)

Series: Derived Chief Lower central Upper central

C1C7 — C7×C7⋊C3
C1C7C72 — C7×C7⋊C3
C7 — C7×C7⋊C3
C1C7

Generators and relations for C7×C7⋊C3
 G = < a,b,c | a7=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
3C7
3C7
7C21

Permutation representations of C7×C7⋊C3
On 21 points - transitive group 21T13
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)

G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,2,3,4,5,6,7),(8,10,12,14,9,11,13),(15,19,16,20,17,21,18)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)])

G:=TransitiveGroup(21,13);

C7×C7⋊C3 is a maximal subgroup of   C75F7

35 conjugacy classes

class 1 3A3B7A···7F7G···7T21A···21L
order1337···77···721···21
size1771···13···37···7

35 irreducible representations

dim111133
type+
imageC1C3C7C21C7⋊C3C7×C7⋊C3
kernelC7×C7⋊C3C72C7⋊C3C7C7C1
# reps12612212

Matrix representation of C7×C7⋊C3 in GL3(𝔽43) generated by

3500
0350
0035
,
4100
040
2516
,
010
73720
006
G:=sub<GL(3,GF(43))| [35,0,0,0,35,0,0,0,35],[41,0,2,0,4,5,0,0,16],[0,7,0,1,37,0,0,20,6] >;

C7×C7⋊C3 in GAP, Magma, Sage, TeX

C_7\times C_7\rtimes C_3
% in TeX

G:=Group("C7xC7:C3");
// GroupNames label

G:=SmallGroup(147,3);
// by ID

G=gap.SmallGroup(147,3);
# by ID

G:=PCGroup([3,-3,-7,-7,380]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C7×C7⋊C3 in TeX

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