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G = C4×C7⋊C3order 84 = 22·3·7

Direct product of C4 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C4×C7⋊C3, C28⋊C3, C72C12, C14.2C6, C2.(C2×C7⋊C3), (C2×C7⋊C3).2C2, SmallGroup(84,2)

Series: Derived Chief Lower central Upper central

C1C7 — C4×C7⋊C3
C1C7C14C2×C7⋊C3 — C4×C7⋊C3
C7 — C4×C7⋊C3
C1C4

Generators and relations for C4×C7⋊C3
 G = < a,b,c | a4=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C6
7C12

Character table of C4×C7⋊C3

 class 123A3B4A4B6A6B7A7B12A12B12C12D14A14B28A28B28C28D
 size 11771177337777333333
ρ111111111111111111111    trivial
ρ21111-1-11111-1-1-1-111-1-1-1-1    linear of order 2
ρ311ζ32ζ311ζ32ζ311ζ3ζ32ζ3ζ32111111    linear of order 3
ρ411ζ32ζ3-1-1ζ32ζ311ζ65ζ6ζ65ζ611-1-1-1-1    linear of order 6
ρ511ζ3ζ3211ζ3ζ3211ζ32ζ3ζ32ζ3111111    linear of order 3
ρ611ζ3ζ32-1-1ζ3ζ3211ζ6ζ65ζ6ζ6511-1-1-1-1    linear of order 6
ρ71-111i-i-1-111-i-iii-1-1i-ii-i    linear of order 4
ρ81-111-ii-1-111ii-i-i-1-1-ii-ii    linear of order 4
ρ91-1ζ32ζ3-iiζ6ζ6511ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32-1-1-ii-ii    linear of order 12
ρ101-1ζ3ζ32-iiζ65ζ611ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3-1-1-ii-ii    linear of order 12
ρ111-1ζ3ζ32i-iζ65ζ611ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3-1-1i-ii-i    linear of order 12
ρ121-1ζ32ζ3i-iζ6ζ6511ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32-1-1i-ii-i    linear of order 12
ρ1333003300-1--7/2-1+-7/20000-1+-7/2-1--7/2-1--7/2-1--7/2-1+-7/2-1+-7/2    complex lifted from C7⋊C3
ρ1433003300-1+-7/2-1--7/20000-1--7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2-1--7/2    complex lifted from C7⋊C3
ρ153300-3-300-1+-7/2-1--7/20000-1--7/2-1+-7/21--7/21--7/21+-7/21+-7/2    complex lifted from C2×C7⋊C3
ρ163300-3-300-1--7/2-1+-7/20000-1+-7/2-1--7/21+-7/21+-7/21--7/21--7/2    complex lifted from C2×C7⋊C3
ρ173-3003i-3i00-1+-7/2-1--7/200001+-7/21--7/2ζ4ζ744ζ724ζ7ζ43ζ7443ζ7243ζ7ζ4ζ764ζ754ζ73ζ43ζ7643ζ7543ζ73    complex faithful
ρ183-300-3i3i00-1+-7/2-1--7/200001+-7/21--7/2ζ43ζ7443ζ7243ζ7ζ4ζ744ζ724ζ7ζ43ζ7643ζ7543ζ73ζ4ζ764ζ754ζ73    complex faithful
ρ193-300-3i3i00-1--7/2-1+-7/200001--7/21+-7/2ζ43ζ7643ζ7543ζ73ζ4ζ764ζ754ζ73ζ43ζ7443ζ7243ζ7ζ4ζ744ζ724ζ7    complex faithful
ρ203-3003i-3i00-1--7/2-1+-7/200001--7/21+-7/2ζ4ζ764ζ754ζ73ζ43ζ7643ζ7543ζ73ζ4ζ744ζ724ζ7ζ43ζ7443ζ7243ζ7    complex faithful

Permutation representations of C4×C7⋊C3
On 28 points - transitive group 28T13
Generators in S28
(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 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)| (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (1,2,3,4,5,6,7)(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( (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (1,2,3,4,5,6,7)(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([(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21)], [(1,2,3,4,5,6,7),(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(28,13);

C4×C7⋊C3 is a maximal subgroup of   C7⋊C24  C4.F7  C4⋊F7  C28.A4

Matrix representation of C4×C7⋊C3 in GL4(𝔽337) generated by

189000
0100
0010
0001
,
1000
03361241
001241
03361251
,
208000
01251213
0100
011212
G:=sub<GL(4,GF(337))| [189,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,336,0,336,0,124,124,125,0,1,1,1],[208,0,0,0,0,125,1,1,0,1,0,1,0,213,0,212] >;

C4×C7⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes C_3
% in TeX

G:=Group("C4xC7:C3");
// GroupNames label

G:=SmallGroup(84,2);
// by ID

G=gap.SmallGroup(84,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-7,24,199]);
// Polycyclic

G:=Group<a,b,c|a^4=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 C4×C7⋊C3 in TeX
Character table of C4×C7⋊C3 in TeX

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