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## G = C32⋊3Dic9order 324 = 22·34

### The semidirect product of C32 and Dic9 acting via Dic9/C9=C4

Aliases: C323Dic9, C33.5Dic3, C3⋊S3.D9, C9⋊(C32⋊C4), (C32×C9)⋊2C4, C3.(C33⋊C4), (C3×C3⋊S3).3S3, (C9×C3⋊S3).2C2, SmallGroup(324,112)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C32⋊3Dic9
 Chief series C1 — C3 — C9 — C32×C9 — C9×C3⋊S3 — C32⋊3Dic9
 Lower central C32×C9 — C32⋊3Dic9
 Upper central C1

Generators and relations for C323Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C323Dic9

 class 1 2 3A 3B 3C 3D 3E 3F 3G 4A 4B 6 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O 18A 18B 18C size 1 9 2 4 4 4 4 4 4 81 81 18 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 18 18 18 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 1 1 trivial ρ2 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 1 1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 1 i -i -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 1 1 1 1 1 1 -i i -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 2 2 2 2 0 0 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ7 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ8 2 2 -1 -1 -1 2 2 -1 -1 0 0 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ9 2 -2 2 2 2 2 2 2 2 0 0 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 -1 -1 -1 2 2 -1 -1 0 0 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ11 2 -2 -1 -1 -1 2 2 -1 -1 0 0 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ12 2 -2 -1 -1 -1 2 2 -1 -1 0 0 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ13 4 0 4 -2 1 -2 1 -2 1 0 0 0 4 4 4 -2 1 -2 -2 1 -2 -2 -2 1 1 1 1 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 0 4 1 -2 1 -2 1 -2 0 0 0 4 4 4 1 -2 1 1 -2 1 1 1 -2 -2 -2 -2 0 0 0 orthogonal lifted from C32⋊C4 ρ15 4 0 4 -2 1 -2 1 -2 1 0 0 0 -2 -2 -2 1 -1+3√-3/2 1 1 -1-3√-3/2 1 1 1 -1-3√-3/2 -1+3√-3/2 -1-3√-3/2 -1+3√-3/2 0 0 0 complex lifted from C33⋊C4 ρ16 4 0 4 1 -2 1 -2 1 -2 0 0 0 -2 -2 -2 -1+3√-3/2 1 -1+3√-3/2 -1+3√-3/2 1 -1-3√-3/2 -1-3√-3/2 -1-3√-3/2 1 1 1 1 0 0 0 complex lifted from C33⋊C4 ρ17 4 0 4 -2 1 -2 1 -2 1 0 0 0 -2 -2 -2 1 -1-3√-3/2 1 1 -1+3√-3/2 1 1 1 -1+3√-3/2 -1-3√-3/2 -1+3√-3/2 -1-3√-3/2 0 0 0 complex lifted from C33⋊C4 ρ18 4 0 4 1 -2 1 -2 1 -2 0 0 0 -2 -2 -2 -1-3√-3/2 1 -1-3√-3/2 -1-3√-3/2 1 -1+3√-3/2 -1+3√-3/2 -1+3√-3/2 1 1 1 1 0 0 0 complex lifted from C33⋊C4 ρ19 4 0 -2 -1+3√-3/2 1 1 -2 -1-3√-3/2 1 0 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95-ζ94 -ζ98-ζ9 2ζ98-ζ9 -ζ97+2ζ92 -ζ97-ζ92 2ζ97-ζ92 -ζ98+2ζ9 -ζ95+2ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ95-ζ94 -ζ97-ζ92 0 0 0 complex faithful ρ20 4 0 -2 -1+3√-3/2 1 1 -2 -1-3√-3/2 1 0 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 -ζ97+2ζ92 -ζ95-ζ94 2ζ95-ζ94 2ζ98-ζ9 -ζ98-ζ9 -ζ98+2ζ9 -ζ95+2ζ94 2ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ97-ζ92 -ζ98-ζ9 0 0 0 complex faithful ρ21 4 0 -2 1 -1+3√-3/2 -2 1 1 -1-3√-3/2 0 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 -ζ98-ζ9 -ζ97+2ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95+2ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 2ζ97-ζ92 2ζ98-ζ9 -ζ98+2ζ9 2ζ95-ζ94 0 0 0 complex faithful ρ22 4 0 -2 -1+3√-3/2 1 1 -2 -1-3√-3/2 1 0 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98-ζ9 -ζ97-ζ92 -ζ97+2ζ92 2ζ95-ζ94 -ζ95-ζ94 -ζ95+2ζ94 2ζ97-ζ92 -ζ98+2ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ98-ζ9 -ζ95-ζ94 0 0 0 complex faithful ρ23 4 0 -2 -1-3√-3/2 1 1 -2 -1+3√-3/2 1 0 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97-ζ92 -ζ95-ζ94 -ζ95+2ζ94 -ζ98+2ζ9 -ζ98-ζ9 2ζ98-ζ9 2ζ95-ζ94 -ζ97+2ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ97-ζ92 -ζ98-ζ9 0 0 0 complex faithful ρ24 4 0 -2 1 -1-3√-3/2 -2 1 1 -1+3√-3/2 0 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 -ζ95-ζ94 -ζ98+2ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97+2ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 2ζ98-ζ9 -ζ95+2ζ94 2ζ95-ζ94 2ζ97-ζ92 0 0 0 complex faithful ρ25 4 0 -2 -1-3√-3/2 1 1 -2 -1+3√-3/2 1 0 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 -ζ98+2ζ9 -ζ97-ζ92 2ζ97-ζ92 -ζ95+2ζ94 -ζ95-ζ94 2ζ95-ζ94 -ζ97+2ζ92 2ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ98-ζ9 -ζ95-ζ94 0 0 0 complex faithful ρ26 4 0 -2 1 -1-3√-3/2 -2 1 1 -1+3√-3/2 0 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 -ζ98-ζ9 2ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 2ζ95-ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ97+2ζ92 -ζ98+2ζ9 2ζ98-ζ9 -ζ95+2ζ94 0 0 0 complex faithful ρ27 4 0 -2 -1-3√-3/2 1 1 -2 -1+3√-3/2 1 0 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 -ζ95+2ζ94 -ζ98-ζ9 -ζ98+2ζ9 2ζ97-ζ92 -ζ97-ζ92 -ζ97+2ζ92 2ζ98-ζ9 2ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ95-ζ94 -ζ97-ζ92 0 0 0 complex faithful ρ28 4 0 -2 1 -1+3√-3/2 -2 1 1 -1-3√-3/2 0 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 -ζ95-ζ94 2ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 2ζ97-ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ98+2ζ9 2ζ95-ζ94 -ζ95+2ζ94 -ζ97+2ζ92 0 0 0 complex faithful ρ29 4 0 -2 1 -1+3√-3/2 -2 1 1 -1-3√-3/2 0 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 -ζ97-ζ92 2ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98+2ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ95+2ζ94 -ζ97+2ζ92 2ζ97-ζ92 2ζ98-ζ9 0 0 0 complex faithful ρ30 4 0 -2 1 -1-3√-3/2 -2 1 1 -1+3√-3/2 0 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 -ζ97-ζ92 -ζ95+2ζ94 -ζ95-ζ94 -ζ98-ζ9 2ζ98-ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 2ζ95-ζ94 2ζ97-ζ92 -ζ97+2ζ92 -ζ98+2ζ9 0 0 0 complex faithful

Smallest permutation representation of C323Dic9
On 36 points
Generators in S36
(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)
(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)
(1 34 10 25)(2 33 11 24)(3 32 12 23)(4 31 13 22)(5 30 14 21)(6 29 15 20)(7 28 16 19)(8 27 17 36)(9 26 18 35)

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

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

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

Matrix representation of C323Dic9 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 26
,
 10 0 0 0 0 26 0 0 0 0 10 0 0 0 0 26
,
 0 34 0 0 34 0 0 0 0 0 0 12 0 0 12 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,26],[10,0,0,0,0,26,0,0,0,0,10,0,0,0,0,26],[0,34,0,0,34,0,0,0,0,0,0,12,0,0,12,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C323Dic9 in GAP, Magma, Sage, TeX

C_3^2\rtimes_3{\rm Dic}_9
% in TeX

G:=Group("C3^2:3Dic9");
// GroupNames label

G:=SmallGroup(324,112);
// by ID

G=gap.SmallGroup(324,112);
# by ID

G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,362,80,387,297,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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