Copied to
clipboard

G = C22×C7⋊C3order 84 = 22·3·7

Direct product of C22 and C7⋊C3

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

Aliases: C22×C7⋊C3, C142C6, C72(C2×C6), (C2×C14)⋊3C3, SmallGroup(84,9)

Series: Derived Chief Lower central Upper central

C1C7 — C22×C7⋊C3
C1C7C7⋊C3C2×C7⋊C3 — C22×C7⋊C3
C7 — C22×C7⋊C3
C1C22

Generators and relations for C22×C7⋊C3
 G = < a,b,c,d | a2=b2=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

7C3
7C6
7C6
7C6
7C2×C6

Character table of C22×C7⋊C3

 class 12A2B2C3A3B6A6B6C6D6E6F7A7B14A14B14C14D14E14F
 size 11117777777733333333
ρ111111111111111111111    trivial
ρ211-1-111-111-1-1-1111-1-1-1-11    linear of order 2
ρ31-1-11111-1-1-11-111-1-111-1-1    linear of order 2
ρ41-11-111-1-1-11-1111-11-1-11-1    linear of order 2
ρ51-1-11ζ3ζ32ζ3ζ6ζ65ζ6ζ32ζ6511-1-111-1-1    linear of order 6
ρ61-11-1ζ32ζ3ζ6ζ65ζ6ζ3ζ65ζ3211-11-1-11-1    linear of order 6
ρ711-1-1ζ3ζ32ζ65ζ32ζ3ζ6ζ6ζ65111-1-1-1-11    linear of order 6
ρ81-1-11ζ32ζ3ζ32ζ65ζ6ζ65ζ3ζ611-1-111-1-1    linear of order 6
ρ91111ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ101111ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ311111111    linear of order 3
ρ1111-1-1ζ32ζ3ζ6ζ3ζ32ζ65ζ65ζ6111-1-1-1-11    linear of order 6
ρ121-11-1ζ3ζ32ζ65ζ6ζ65ζ32ζ6ζ311-11-1-11-1    linear of order 6
ρ133-3-3300000000-1+-7/2-1--7/21--7/21+-7/2-1--7/2-1+-7/21--7/21+-7/2    complex lifted from C2×C7⋊C3
ρ143-33-300000000-1+-7/2-1--7/21--7/2-1--7/21+-7/21--7/2-1+-7/21+-7/2    complex lifted from C2×C7⋊C3
ρ15333300000000-1+-7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ1633-3-300000000-1+-7/2-1--7/2-1+-7/21+-7/21+-7/21--7/21--7/2-1--7/2    complex lifted from C2×C7⋊C3
ρ17333300000000-1--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ183-3-3300000000-1--7/2-1+-7/21+-7/21--7/2-1+-7/2-1--7/21+-7/21--7/2    complex lifted from C2×C7⋊C3
ρ1933-3-300000000-1--7/2-1+-7/2-1--7/21--7/21--7/21+-7/21+-7/2-1+-7/2    complex lifted from C2×C7⋊C3
ρ203-33-300000000-1--7/2-1+-7/21+-7/2-1+-7/21--7/21+-7/2-1--7/21--7/2    complex lifted from C2×C7⋊C3

Permutation representations of C22×C7⋊C3
On 28 points - transitive group 28T14
Generators in S28
(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)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)

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

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

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

G:=TransitiveGroup(28,14);

C22×C7⋊C3 is a maximal subgroup of   Dic7⋊C6

Matrix representation of C22×C7⋊C3 in GL4(𝔽43) generated by

42000
04200
00420
00042
,
1000
04200
00420
00042
,
1000
024251
0100
0010
,
6000
0100
0184242
0010
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,24,1,0,0,25,0,1,0,1,0,0],[6,0,0,0,0,1,18,0,0,0,42,1,0,0,42,0] >;

C22×C7⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes C_3
% in TeX

G:=Group("C2^2xC7:C3");
// GroupNames label

G:=SmallGroup(84,9);
// by ID

G=gap.SmallGroup(84,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,107]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

Export

Subgroup lattice of C22×C7⋊C3 in TeX
Character table of C22×C7⋊C3 in TeX

׿
×
𝔽