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G = C32⋊C12order 108 = 22·33

The semidirect product of C32 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: C32⋊C12, He32C4, C321Dic3, (C3×C6).C6, C3⋊Dic3⋊C3, C6.2(C3×S3), (C3×C6).1S3, C2.(C32⋊C6), (C2×He3).1C2, C3.2(C3×Dic3), SmallGroup(108,8)

Series: Derived Chief Lower central Upper central

C1C32 — C32⋊C12
C1C3C32C3×C6C2×He3 — C32⋊C12
C32 — C32⋊C12
C1C2

Generators and relations for C32⋊C12
 G = < a,b,c | a3=b3=c12=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >

3C3
3C3
6C3
9C4
3C6
3C6
6C6
2C32
3Dic3
9C12
9Dic3
2C3×C6
3C3×Dic3

Character table of C32⋊C12

 class 123A3B3C3D3E3F4A4B6A6B6C6D6E6F12A12B12C12D
 size 11233666992336669999
ρ111111111111111111111    trivial
ρ211111111-1-1111111-1-1-1-1    linear of order 2
ρ3111ζ3ζ32ζ31ζ32111ζ3ζ321ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ4111ζ32ζ3ζ321ζ3111ζ32ζ31ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ5111ζ32ζ3ζ321ζ3-1-11ζ32ζ31ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ6111ζ3ζ32ζ31ζ32-1-11ζ3ζ321ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ71-1111111-ii-1-1-1-1-1-1i-ii-i    linear of order 4
ρ81-1111111i-i-1-1-1-1-1-1-ii-ii    linear of order 4
ρ91-11ζ32ζ3ζ321ζ3i-i-1ζ6ζ65-1ζ65ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ101-11ζ3ζ32ζ31ζ32i-i-1ζ65ζ6-1ζ6ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ111-11ζ3ζ32ζ31ζ32-ii-1ζ65ζ6-1ζ6ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ121-11ζ32ζ3ζ321ζ3-ii-1ζ6ζ65-1ζ65ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ1322222-1-1-100222-1-1-10000    orthogonal lifted from S3
ρ142-2222-1-1-100-2-2-21110000    symplectic lifted from Dic3, Schur index 2
ρ152-22-1+-3-1--3ζ65-1ζ600-21--31+-31ζ32ζ30000    complex lifted from C3×Dic3
ρ162-22-1--3-1+-3ζ6-1ζ6500-21+-31--31ζ3ζ320000    complex lifted from C3×Dic3
ρ17222-1+-3-1--3ζ65-1ζ6002-1+-3-1--3-1ζ6ζ650000    complex lifted from C3×S3
ρ18222-1--3-1+-3ζ6-1ζ65002-1--3-1+-3-1ζ65ζ60000    complex lifted from C3×S3
ρ1966-30000000-3000000000    orthogonal lifted from C32⋊C6
ρ206-6-300000003000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊C12
On 36 points
Generators in S36
(1 28 5)(2 25 10)(3 21 17)(4 8 31)(6 20 24)(7 34 11)(9 15 23)(12 14 18)(13 30 26)(16 29 33)(19 36 32)(22 35 27)
(1 32 23)(2 24 33)(3 34 13)(4 14 35)(5 36 15)(6 16 25)(7 26 17)(8 18 27)(9 28 19)(10 20 29)(11 30 21)(12 22 31)
(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 29 30 31 32 33 34 35 36)

G:=sub<Sym(36)| (1,28,5)(2,25,10)(3,21,17)(4,8,31)(6,20,24)(7,34,11)(9,15,23)(12,14,18)(13,30,26)(16,29,33)(19,36,32)(22,35,27), (1,32,23)(2,24,33)(3,34,13)(4,14,35)(5,36,15)(6,16,25)(7,26,17)(8,18,27)(9,28,19)(10,20,29)(11,30,21)(12,22,31), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)>;

G:=Group( (1,28,5)(2,25,10)(3,21,17)(4,8,31)(6,20,24)(7,34,11)(9,15,23)(12,14,18)(13,30,26)(16,29,33)(19,36,32)(22,35,27), (1,32,23)(2,24,33)(3,34,13)(4,14,35)(5,36,15)(6,16,25)(7,26,17)(8,18,27)(9,28,19)(10,20,29)(11,30,21)(12,22,31), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36) );

G=PermutationGroup([(1,28,5),(2,25,10),(3,21,17),(4,8,31),(6,20,24),(7,34,11),(9,15,23),(12,14,18),(13,30,26),(16,29,33),(19,36,32),(22,35,27)], [(1,32,23),(2,24,33),(3,34,13),(4,14,35),(5,36,15),(6,16,25),(7,26,17),(8,18,27),(9,28,19),(10,20,29),(11,30,21),(12,22,31)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)])

C32⋊C12 is a maximal subgroup of
He32Q8  C6.S32  He3⋊(C2×C4)  He33D4  He33Q8  C4×C32⋊C6  He36D4  C33⋊C12  He3.Dic3  He3.2Dic3  C33⋊Dic3  He3.3Dic3  He3⋊Dic3  C334C12  He3.4Dic3  C32⋊CSU2(𝔽3)  C625Dic3  C6.(S3×A4)  C624C12
C32⋊C12 is a maximal quotient of
He33C8  C32⋊C36  C32⋊Dic9  He3⋊C12  C33⋊C12  He3.C12  He3.Dic3  He3.2C12  He3.2Dic3  C334C12  C625Dic3  C624C12

Matrix representation of C32⋊C12 in GL6(𝔽13)

001000
000100
000010
000001
100000
010000
,
1210000
1200000
0012100
0012000
0000121
0000120
,
12712777
616116
712127712
666166
7127777
661616

G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,6,7,6,7,6,7,1,12,6,12,6,12,6,12,6,7,1,7,1,7,1,7,6,7,1,7,6,7,1,7,6,12,6,7,6] >;

C32⋊C12 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_{12}
% in TeX

G:=Group("C3^2:C12");
// GroupNames label

G:=SmallGroup(108,8);
// by ID

G=gap.SmallGroup(108,8);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,483,488,1804]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊C12 in TeX
Character table of C32⋊C12 in TeX

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