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G = C323Dic9order 324 = 22·34

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

metabelian, soluble, monomial, A-group

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

C1C32×C9 — C323Dic9
C1C3C9C32×C9C9×C3⋊S3 — C323Dic9
C32×C9 — C323Dic9
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 >

9C2
2C3
2C3
4C3
4C3
81C4
6S3
6S3
9C6
2C32
2C32
4C32
4C32
4C9
4C9
27Dic3
6C3×S3
6C3×S3
9C18
2C3×C9
2C3×C9
4C3×C9
4C3×C9
9Dic9
9C32⋊C4
6S3×C9
6S3×C9
3C33⋊C4

Character table of C323Dic9

 class 123A3B3C3D3E3F3G4A4B69A9B9C9D9E9F9G9H9I9J9K9L9M9N9O18A18B18C
 size 192444444818118222444444444444181818
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-11111111111111111111    linear of order 2
ρ31-11111111i-i-1111111111111111-1-1-1    linear of order 4
ρ41-11111111-ii-1111111111111111-1-1-1    linear of order 4
ρ5222222222002-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-1-1-122-1-100-1ζ989ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ989ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ722-1-1-122-1-100-1ζ9792ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9792ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ822-1-1-122-1-100-1ζ9594ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ9594ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ92-2222222200-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1111    symplectic lifted from Dic3, Schur index 2
ρ102-2-1-1-122-1-1001ζ989ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ989ζ959498997929594    symplectic lifted from Dic9, Schur index 2
ρ112-2-1-1-122-1-1001ζ9594ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ9594ζ979295949899792    symplectic lifted from Dic9, Schur index 2
ρ122-2-1-1-122-1-1001ζ9792ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9792ζ98997929594989    symplectic lifted from Dic9, Schur index 2
ρ13404-21-21-21000444-21-2-21-2-2-21111000    orthogonal lifted from C32⋊C4
ρ144041-21-21-20004441-211-2111-2-2-2-2000    orthogonal lifted from C32⋊C4
ρ15404-21-21-21000-2-2-21-1+3-3/211-1-3-3/2111-1-3-3/2-1+3-3/2-1-3-3/2-1+3-3/2000    complex lifted from C33⋊C4
ρ164041-21-21-2000-2-2-2-1+3-3/21-1+3-3/2-1+3-3/21-1-3-3/2-1-3-3/2-1-3-3/21111000    complex lifted from C33⋊C4
ρ17404-21-21-21000-2-2-21-1-3-3/211-1+3-3/2111-1+3-3/2-1-3-3/2-1+3-3/2-1-3-3/2000    complex lifted from C33⋊C4
ρ184041-21-21-2000-2-2-2-1-3-3/21-1-3-3/2-1-3-3/21-1+3-3/2-1+3-3/2-1+3-3/21111000    complex lifted from C33⋊C4
ρ1940-2-1+3-3/211-2-1-3-3/2100095+2ζ9497+2ζ9298+2ζ9959498998997+2ζ929792979298+2ζ995+2ζ94989959495949792000    complex faithful
ρ2040-2-1+3-3/211-2-1-3-3/2100097+2ζ9298+2ζ995+2ζ9497+2ζ929594959498998998+2ζ995+2ζ949792959497929792989000    complex faithful
ρ2140-21-1+3-3/2-211-1-3-3/200098+2ζ995+2ζ9497+2ζ9298997+2ζ929792959495+2ζ9495949792989979298998+2ζ99594000    complex faithful
ρ2240-2-1+3-3/211-2-1-3-3/2100098+2ζ995+2ζ9497+2ζ92989979297+2ζ929594959495+2ζ94979298+2ζ997929899899594000    complex faithful
ρ2340-2-1-3-3/211-2-1+3-3/2100097+2ζ9298+2ζ995+2ζ949792959495+2ζ9498+2ζ9989989959497+2ζ92959497929792989000    complex faithful
ρ2440-21-1-3-3/2-211-1+3-3/200095+2ζ9497+2ζ9298+2ζ9959498+2ζ9989979297+2ζ929792989959498995+2ζ9495949792000    complex faithful
ρ2540-2-1-3-3/211-2-1+3-3/2100098+2ζ995+2ζ9497+2ζ9298+2ζ99792979295+2ζ949594959497+2ζ9298997929899899594000    complex faithful
ρ2640-21-1-3-3/2-211-1+3-3/200098+2ζ995+2ζ9497+2ζ9298997929792959495949594979298997+2ζ9298+2ζ998995+2ζ94000    complex faithful
ρ2740-2-1-3-3/211-2-1+3-3/2100095+2ζ9497+2ζ9298+2ζ995+2ζ9498998+2ζ99792979297+2ζ929899594989959495949792000    complex faithful
ρ2840-21-1+3-3/2-211-1-3-3/200095+2ζ9497+2ζ9298+2ζ99594989989979297929792989959498+2ζ9959495+2ζ9497+2ζ92000    complex faithful
ρ2940-21-1+3-3/2-211-1-3-3/200097+2ζ9298+2ζ995+2ζ9497929594959498998+2ζ99899594979295+2ζ9497+2ζ929792989000    complex faithful
ρ3040-21-1-3-3/2-211-1+3-3/200097+2ζ9298+2ζ995+2ζ94979295+2ζ949594989989989959497929594979297+2ζ9298+2ζ9000    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

1000
0100
00100
00026
,
10000
02600
00100
00026
,
03400
34000
00012
00120
,
0010
0001
0100
1000
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

Export

Subgroup lattice of C323Dic9 in TeX
Character table of C323Dic9 in TeX

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