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## G = C62.6C32order 324 = 22·34

### 6th non-split extension by C62 of C32 acting faithfully

Aliases: C62.6C32, 3- 1+22A4, C222C3≀C3, C32⋊A43C3, (C32×A4)⋊2C3, C32.A45C3, C32.6(C3×A4), (C2×C6).10He3, C3.11(C32⋊A4), (C22×3- 1+2)⋊2C3, SmallGroup(324,58)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.6C32
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — C62.6C32
 Lower central C22 — C2×C6 — C62 — C62.6C32
 Upper central C1 — C3 — C32 — 3- 1+2

Generators and relations for C62.6C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab3, dad-1=ab2, cbc-1=a3b4, bd=db, dcd-1=a2b4c >

3C2
3C3
12C3
12C3
12C3
36C3
3C6
9C6
3C9
12C32
12C32
12C32
12C9
12C32
12C32
3A4
3A4
3A4
9A4
3C18
3C18
3C18
4He3
4C33

Character table of C62.6C32

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 6A 6B 6C 6D 9A 9B 9C 9D 18A 18B 18C 18D 18E 18F size 1 3 1 1 3 3 12 12 12 12 12 12 36 36 3 3 9 9 9 9 36 36 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ10 3 -1 3 3 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 3 3 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 3 -3-3√-3/2 -3+3√-3/2 0 0 3+√-3/2 3-√-3/2 √-3 -3+√-3/2 -√-3 -3-√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ12 3 -1 3 3 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 complex lifted from C3×A4 ρ13 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ6 ζ65 0 0 0 0 2 -1+√-3 -1+√-3 -1-√-3 -1-√-3 2 complex lifted from C32⋊A4 ρ14 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -3+√-3/2 -3-√-3/2 3-√-3/2 -√-3 3+√-3/2 √-3 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ15 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ65 ζ6 0 0 0 0 -1+√-3 2 -1+√-3 2 -1-√-3 -1-√-3 complex lifted from C32⋊A4 ρ16 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ65 ζ6 0 0 0 0 -1-√-3 -1+√-3 2 -1-√-3 2 -1+√-3 complex lifted from C32⋊A4 ρ17 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ18 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 complex lifted from He3 ρ19 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ6 ζ65 0 0 0 0 -1-√-3 2 -1-√-3 2 -1+√-3 -1+√-3 complex lifted from C32⋊A4 ρ20 3 -1 3 3 3 3 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 complex lifted from C3×A4 ρ21 3 3 -3+3√-3/2 -3-3√-3/2 0 0 √-3 -√-3 -3+√-3/2 3-√-3/2 -3-√-3/2 3+√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ22 3 3 -3+3√-3/2 -3-3√-3/2 0 0 3-√-3/2 3+√-3/2 -√-3 -3-√-3/2 √-3 -3+√-3/2 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ23 3 3 -3+3√-3/2 -3-3√-3/2 0 0 -3-√-3/2 -3+√-3/2 3+√-3/2 √-3 3-√-3/2 -√-3 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ24 3 -1 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ65 ζ6 0 0 0 0 2 -1-√-3 -1-√-3 -1+√-3 -1+√-3 2 complex lifted from C32⋊A4 ρ25 3 3 -3-3√-3/2 -3+3√-3/2 0 0 -√-3 √-3 -3-√-3/2 3+√-3/2 -3+√-3/2 3-√-3/2 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3≀C3 ρ26 3 -1 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1 -1 ζ6 ζ65 0 0 0 0 -1+√-3 -1-√-3 2 -1+√-3 2 -1-√-3 complex lifted from C32⋊A4 ρ27 9 -3 -9+9√-3/2 -9-9√-3/2 0 0 0 0 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ28 9 -3 -9-9√-3/2 -9+9√-3/2 0 0 0 0 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C62.6C32
On 36 points
Generators in S36
(1 23)(2 27 8 24 5 21)(3 22 6 25 9 19)(4 26)(7 20)(10 33 16 30 13 36)(11 28 14 31 17 34)(12 32)(15 35)(18 29)
(1 18 4 12 7 15)(2 10 5 13 8 16)(3 11 6 14 9 17)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)
(2 8 5)(10 36 24)(11 31 22)(12 32 23)(13 30 27)(14 34 25)(15 35 26)(16 33 21)(17 28 19)(18 29 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 25 26 27)(28 29 30 31 32 33 34 35 36)

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

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

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

Matrix representation of C62.6C32 in GL6(𝔽19)

 0 18 1 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 0 1 0 0 0 0 0 12 7 0 0 0 0 1 0 11
,
 0 1 18 0 0 0 1 0 18 0 0 0 0 0 18 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 18 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 12 0 7
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 10 0 0 0 0 0 18 1 0 0 0 1 18 0

G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,18,18,18,0,0,0,1,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,7,0,0,0,0,0,0,11],[0,1,0,0,0,0,1,0,0,0,0,0,18,18,18,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,18,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,1,0,0,0,10,18,18,0,0,0,0,1,0] >;

C62.6C32 in GAP, Magma, Sage, TeX

C_6^2._6C_3^2
% in TeX

G:=Group("C6^2.6C3^2");
// GroupNames label

G:=SmallGroup(324,58);
// by ID

G=gap.SmallGroup(324,58);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,115,224,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=b^2,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a*b^2,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^4*c>;
// generators/relations

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