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G = C34⋊C4order 324 = 22·34

3rd semidirect product of C34 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C343C4, C338Dic3, C3⋊(C33⋊C4), C323(C32⋊C4), C323(C3⋊Dic3), C3⋊S3.(C3⋊S3), (C3×C3⋊S3).6S3, (C32×C3⋊S3).3C2, SmallGroup(324,163)

Series: Derived Chief Lower central Upper central

C1C34 — C34⋊C4
C1C3C32C34C32×C3⋊S3 — C34⋊C4
C34 — C34⋊C4
C1

Generators and relations for C34⋊C4
 G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a-1b, bc=cb, bd=db, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 564 in 104 conjugacy classes, 19 normal (7 characteristic)
C1, C2, C3 [×4], C3 [×10], C4, S3 [×2], C6 [×4], C32 [×2], C32 [×36], Dic3 [×4], C3×S3 [×8], C3⋊S3, C3×C6, C33 [×4], C33 [×10], C3⋊Dic3, C32⋊C4, S3×C32 [×2], C3×C3⋊S3 [×4], C34, C33⋊C4 [×4], C32×C3⋊S3, C34⋊C4
Quotients: C1, C2, C4, S3 [×4], Dic3 [×4], C3⋊S3, C3⋊Dic3, C32⋊C4, C33⋊C4 [×4], C34⋊C4

Character table of C34⋊C4

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T3U3V4A4B6A6B6C6D
 size 192222444444444444444444818118181818
ρ1111111111111111111111111111111    trivial
ρ2111111111111111111111111-1-11111    linear of order 2
ρ31-11111111111111111111111i-i-1-1-1-1    linear of order 4
ρ41-11111111111111111111111-ii-1-1-1-1    linear of order 4
ρ522-12-1-1-1-1-12-1-1-1222-1-12-1-1-12-100-1-1-12    orthogonal lifted from S3
ρ6222-1-1-1-1-12-1-12-1-122-12-1-12-1-1-100-1-12-1    orthogonal lifted from S3
ρ722-1-1-1222-1-1-1-1-1-1222-1-1-1-1-1-1200-12-1-1    orthogonal lifted from S3
ρ822-1-12-1-1-1-1-12-12-122-1-1-12-12-1-1002-1-1-1    orthogonal lifted from S3
ρ92-2-12-1-1-1-1-12-1-1-1222-1-12-1-1-12-100111-2    symplectic lifted from Dic3, Schur index 2
ρ102-2-1-1-1222-1-1-1-1-1-1222-1-1-1-1-1-12001-211    symplectic lifted from Dic3, Schur index 2
ρ112-22-1-1-1-1-12-1-12-1-122-12-1-12-1-1-10011-21    symplectic lifted from Dic3, Schur index 2
ρ122-2-1-12-1-1-1-1-12-12-122-1-1-12-12-1-100-2111    symplectic lifted from Dic3, Schur index 2
ρ134044441-2-2-2-2-2-2-21-21111111-2000000    orthogonal lifted from C32⋊C4
ρ14404444-21111111-21-2-2-2-2-2-2-21000000    orthogonal lifted from C32⋊C4
ρ1540-24-2-21-1+3-3/2-1+3-3/21-1-3-3/2-1-3-3/2-1+3-3/21-2111-2111-2-1-3-3/2000000    complex lifted from C33⋊C4
ρ1640-24-2-2-1+3-3/211-2111-21-2-1-3-3/2-1-3-3/21-1+3-3/2-1+3-3/2-1-3-3/211000000    complex lifted from C33⋊C4
ρ17404-2-2-2-1-3-3/21-211-2111-2-1+3-3/21-1-3-3/2-1+3-3/21-1-3-3/2-1+3-3/21000000    complex lifted from C33⋊C4
ρ18404-2-2-21-1-3-3/21-1+3-3/2-1-3-3/21-1+3-3/2-1-3-3/2-211-211-211-1+3-3/2000000    complex lifted from C33⋊C4
ρ1940-2-2-241-21111111-21-1-3-3/2-1-3-3/2-1-3-3/2-1+3-3/2-1+3-3/2-1+3-3/2-2000000    complex lifted from C33⋊C4
ρ2040-2-24-2-1-3-3/2111-21-211-2-1+3-3/2-1-3-3/2-1+3-3/21-1+3-3/21-1-3-3/21000000    complex lifted from C33⋊C4
ρ21404-2-2-21-1+3-3/21-1-3-3/2-1+3-3/21-1-3-3/2-1+3-3/2-211-211-211-1-3-3/2000000    complex lifted from C33⋊C4
ρ2240-2-24-2-1+3-3/2111-21-211-2-1-3-3/2-1+3-3/2-1-3-3/21-1-3-3/21-1+3-3/21000000    complex lifted from C33⋊C4
ρ2340-2-24-21-1-3-3/2-1+3-3/2-1-3-3/21-1-3-3/21-1+3-3/2-21111-21-21-1+3-3/2000000    complex lifted from C33⋊C4
ρ2440-24-2-21-1-3-3/2-1-3-3/21-1+3-3/2-1+3-3/2-1-3-3/21-2111-2111-2-1+3-3/2000000    complex lifted from C33⋊C4
ρ2540-2-24-21-1+3-3/2-1-3-3/2-1+3-3/21-1+3-3/21-1-3-3/2-21111-21-21-1-3-3/2000000    complex lifted from C33⋊C4
ρ26404-2-2-2-1+3-3/21-211-2111-2-1-3-3/21-1+3-3/2-1-3-3/21-1+3-3/2-1-3-3/21000000    complex lifted from C33⋊C4
ρ2740-24-2-2-1-3-3/211-2111-21-2-1+3-3/2-1+3-3/21-1-3-3/2-1-3-3/2-1+3-3/211000000    complex lifted from C33⋊C4
ρ2840-2-2-24-21-1+3-3/2-1+3-3/2-1+3-3/2-1-3-3/2-1-3-3/2-1-3-3/2-21-21111111000000    complex lifted from C33⋊C4
ρ2940-2-2-241-21111111-21-1+3-3/2-1+3-3/2-1+3-3/2-1-3-3/2-1-3-3/2-1-3-3/2-2000000    complex lifted from C33⋊C4
ρ3040-2-2-24-21-1-3-3/2-1-3-3/2-1-3-3/2-1+3-3/2-1+3-3/2-1+3-3/2-21-21111111000000    complex lifted from C33⋊C4

Smallest permutation representation of C34⋊C4
On 36 points
Generators in S36
(1 7 16)(2 19 34)(3 14 5)(4 36 17)(6 26 32)(8 30 28)(9 18 29)(10 22 13)(11 31 20)(12 15 24)(21 33 27)(23 25 35)
(1 29 21)(2 10 28)(3 23 31)(4 26 12)(5 35 11)(6 24 17)(7 9 33)(8 19 22)(13 30 34)(14 25 20)(15 36 32)(16 18 27)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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)| (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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( (1,7,16)(2,19,34)(3,14,5)(4,36,17)(6,26,32)(8,30,28)(9,18,29)(10,22,13)(11,31,20)(12,15,24)(21,33,27)(23,25,35), (1,29,21)(2,10,28)(3,23,31)(4,26,12)(5,35,11)(6,24,17)(7,9,33)(8,19,22)(13,30,34)(14,25,20)(15,36,32)(16,18,27), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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([(1,7,16),(2,19,34),(3,14,5),(4,36,17),(6,26,32),(8,30,28),(9,18,29),(10,22,13),(11,31,20),(12,15,24),(21,33,27),(23,25,35)], [(1,29,21),(2,10,28),(3,23,31),(4,26,12),(5,35,11),(6,24,17),(7,9,33),(8,19,22),(13,30,34),(14,25,20),(15,36,32),(16,18,27)], [(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,30,22),(3,23,31),(4,32,24),(5,35,11),(6,12,36),(7,33,9),(8,10,34),(13,19,28),(14,25,20),(15,17,26),(16,27,18)], [(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 C34⋊C4 in GL6(𝔽13)

100000
010000
009000
003300
006010
007001
,
100000
010000
009000
003300
000090
002003
,
100000
010000
003000
000300
002090
0011009
,
12120000
100000
003000
000300
002090
0011009
,
500000
880000
005020
000011
000180
000050

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,6,7,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,2,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,2,11,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,3,0,2,11,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,2,1,8,5,0,0,0,1,0,0] >;

C34⋊C4 in GAP, Magma, Sage, TeX

C_3^4\rtimes C_4
% in TeX

G:=Group("C3^4:C4");
// GroupNames label

G:=SmallGroup(324,163);
// by ID

G=gap.SmallGroup(324,163);
# by ID

G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,506,80,771,297,2164,7781]);
// Polycyclic

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

Export

Character table of C34⋊C4 in TeX

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