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G = C26⋊C3order 192 = 26·3

3rd semidirect product of C26 and C3 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C263C3, C248A4, C22⋊(C22⋊A4), SmallGroup(192,1541)

Series: Derived Chief Lower central Upper central

C1C26 — C26⋊C3
C1C22C24C26 — C26⋊C3
C26 — C26⋊C3
C1

Generators and relations for C26⋊C3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, gag-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, gbg-1=a, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >

Subgroups: 3310 in 1015 conjugacy classes, 45 normal (3 characteristic)
C1, C2 [×21], C3, C22 [×21], C22 [×210], C23 [×465], A4 [×21], C24 [×21], C24 [×210], C25 [×21], C22⋊A4 [×21], C26, C26⋊C3
Quotients: C1, C3, A4 [×21], C22⋊A4 [×21], C26⋊C3

Character table of C26⋊C3

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O2P2Q2R2S2T2U3A3B
 size 13333333333333333333336464
ρ1111111111111111111111111    trivial
ρ21111111111111111111111ζ3ζ32    linear of order 3
ρ31111111111111111111111ζ32ζ3    linear of order 3
ρ43-13-1-1-1-1-1-13-1-1-13-1-1-13-1-1-1300    orthogonal lifted from A4
ρ533-1-1-13-13-1-1-1-1-1-13-1-1-1-1-1-1300    orthogonal lifted from A4
ρ63-1-1-1-13-1-13-1-13-1-1-1-1-133-1-1-100    orthogonal lifted from A4
ρ73-1-13-1-1-1-13-1-1-1-133-1-1-1-13-1-100    orthogonal lifted from A4
ρ8333-1-1-1-1-13-1-1-1-1-1-1-13-1-1-13-100    orthogonal lifted from A4
ρ93-1-1-1-1-1-1-1-1-1-1-13-1-1-1-1-1333300    orthogonal lifted from A4
ρ1033-1-13-13-1-1-1-1-1-1-1-1-1-13-13-1-100    orthogonal lifted from A4
ρ113-13-1-1-13-1-1-13-1-1-13-1-1-13-1-1-100    orthogonal lifted from A4
ρ123-1-1-13-1-1-13-13-1-1-1-13-1-1-1-1-1300    orthogonal lifted from A4
ρ133-1-1-1-1-1-1-1-1-1-1-13-13333-1-1-1-100    orthogonal lifted from A4
ρ1433-13-1-1-1-1-13-1-1-1-1-13-1-13-1-1-100    orthogonal lifted from A4
ρ153-13-1-1-1-13-1-1-13-1-1-13-1-1-13-1-100    orthogonal lifted from A4
ρ163-1-1-1-133-1-1-1-1-1-13-13-1-1-1-13-100    orthogonal lifted from A4
ρ173-1-13-1-1-13-1-13-1-1-1-1-1-13-1-13-100    orthogonal lifted from A4
ρ183-1-1-13-1-1-1-13-13-1-13-1-1-1-1-13-100    orthogonal lifted from A4
ρ193-1-1-1-1-13333-1-13-1-1-1-1-1-1-1-1-100    orthogonal lifted from A4
ρ203-1-13-1-13-1-1-1-13-1-1-1-13-1-1-1-1300    orthogonal lifted from A4
ρ213-1-1-13-1-13-1-1-1-1-13-1-13-13-1-1-100    orthogonal lifted from A4
ρ223-1-1-1-13-1-1-133-1-1-1-1-13-1-13-1-100    orthogonal lifted from A4
ρ233-13333-1-1-1-1-1-13-1-1-1-1-1-1-1-1-100    orthogonal lifted from A4
ρ2433-1-1-1-1-1-1-1-13333-1-1-1-1-1-1-1-100    orthogonal lifted from A4

Permutation representations of C26⋊C3
On 24 points - transitive group 24T390
Generators in S24
(1 14)(2 17)(3 7)(4 19)(5 10)(6 22)(8 16)(9 15)(11 21)(12 23)(13 18)(20 24)
(1 8)(2 15)(3 18)(4 23)(5 20)(6 11)(7 13)(9 17)(10 24)(12 19)(14 16)(21 22)
(1 14)(2 24)(3 11)(4 19)(5 9)(6 18)(7 21)(8 16)(10 15)(12 23)(13 22)(17 20)
(1 12)(2 15)(3 22)(4 16)(5 20)(6 7)(8 19)(9 17)(10 24)(11 13)(14 23)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(1 23)(2 24)(4 8)(5 9)(10 15)(12 14)(16 19)(17 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

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

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

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

G:=TransitiveGroup(24,390);

Matrix representation of C26⋊C3 in GL9(𝔽7)

600000000
010000000
056000000
000100000
000010000
000001000
000000100
000000060
000000006
,
100000000
060000000
306000000
000100000
000010000
000001000
000000600
000000060
000000001
,
100000000
060000000
306000000
000600000
000010000
000006000
000000600
000000010
000000006
,
600000000
060000000
421000000
000100000
000060000
000006000
000000100
000000060
000000006
,
600000000
060000000
421000000
000600000
000060000
000001000
000000600
000000060
000000001
,
600000000
010000000
056000000
000600000
000010000
000006000
000000600
000000010
000000006
,
010000000
355000000
002000000
000010000
000001000
000100000
000000010
000000001
000000100

G:=sub<GL(9,GF(7))| [6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[0,3,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

C26⋊C3 in GAP, Magma, Sage, TeX

C_2^6\rtimes C_3
% in TeX

G:=Group("C2^6:C3");
// GroupNames label

G:=SmallGroup(192,1541);
// by ID

G=gap.SmallGroup(192,1541);
# by ID

G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,191,675,1264,4037,7062]);
// Polycyclic

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

Export

Character table of C26⋊C3 in TeX

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