C3^2.3S4 - GroupNames

G = C₃².₃S₄ order 216 = 2³·3³

2^nd non-split extension by C₃² of S₄ acting via S₄/A₄=C₂

Aliases: C₃².₃S₄, C₆².₇S₃, C₃⋊(C₃.S₄), (C₂×C₆)⋊₂D₉, C₃.A₄⋊₂S₃, C₃.₂(C₃⋊S₄), C₂²⋊₂(C₉⋊S₃), (C₃×C₃.A₄)⋊₃C₂, (C₂×C₆).₃(C₃⋊S₃), SmallGroup(216,94)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₃.A₄ — C₃².₃S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₆² — C₃×C₃.A₄ — C₃².₃S₄

Lower central

C₃×C₃.A₄ — C₃².₃S₄

Upper central

C₁

Generators and relations for C₃².₃S₄
G = < a,b,c,d | a⁶=b⁶=d²=1, c³=b², ab=ba, cac^-1=a⁴b³, dad=a²b³, cbc^-1=a³b, dbd=a³b², dcd=b⁴c² >

Subgroups: 430 in 58 conjugacy classes, 17 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, D₄, C₉, C₃², Dic₃, D₆, C₂×C₆, C₂×C₆, D₉, C₃⋊S₃, C₃×C₆, C₃⋊D₄, C₃×C₉, C₃.A₄, C₃⋊Dic₃, C₂×C₃⋊S₃, C₆², C₉⋊S₃, C₃.S₄, C₃²⋊₇D₄, C₃×C₃.A₄, C₃².₃S₄
Quotients: C₁, C₂, S₃, D₉, C₃⋊S₃, S₄, C₉⋊S₃, C₃.S₄, C₃⋊S₄, C₃².₃S₄

Character table of C₃².₃S₄

class	1	2A	2B	3A	3B	3C	3D	4	6A	6B	6C	6D	9A	9B	9C	9D	9E	9F	9G	9H	9I
size	1	3	54	2	2	2	2	54	6	6	6	6	8	8	8	8	8	8	8	8	8
ρ₁	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	linear of order 2
ρ₃	2	2	0	2	2	2	2	0	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	2	0	-1	-1	-1	2	0	-1	-1	2	-1	2	-1	-1	-1	-1	-1	-1	2	2	orthogonal lifted from S₃
ρ₅	2	2	0	-1	-1	-1	2	0	-1	-1	2	-1	-1	-1	-1	-1	2	2	2	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	0	-1	-1	-1	2	0	-1	-1	2	-1	-1	2	2	2	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	2	0	-1	2	-1	-1	0	-1	-1	-1	2	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₈	2	2	0	-1	2	-1	-1	0	-1	-1	-1	2	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₉	2	2	0	2	-1	-1	-1	0	2	-1	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₀	2	2	0	-1	-1	2	-1	0	-1	2	-1	-1	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₁	2	2	0	-1	-1	2	-1	0	-1	2	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₂	2	2	0	-1	-1	2	-1	0	-1	2	-1	-1	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₃	2	2	0	2	-1	-1	-1	0	2	-1	-1	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₄	2	2	0	2	-1	-1	-1	0	2	-1	-1	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₅	2	2	0	-1	2	-1	-1	0	-1	-1	-1	2	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₆	3	-1	-1	3	3	3	3	1	-1	-1	-1	-1	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₄
ρ₁₇	3	-1	1	3	3	3	3	-1	-1	-1	-1	-1	0	0	0	0	0	0	0	0	0	orthogonal lifted from S₄
ρ₁₈	6	-2	0	-3	-3	-3	6	0	1	1	-2	1	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃⋊S₄
ρ₁₉	6	-2	0	6	-3	-3	-3	0	-2	1	1	1	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃.S₄
ρ₂₀	6	-2	0	-3	6	-3	-3	0	1	1	1	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃.S₄
ρ₂₁	6	-2	0	-3	-3	6	-3	0	1	-2	1	1	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃.S₄

Smallest permutation representation of C₃².₃S₄
►On 54 points

Generators in S₅₄

(1 18 46 32 38 23)(2 10 47 33 39 24)(3 40 48)(4 12 49 35 41 26)(5 13 50 36 42 27)(6 43 51)(7 15 52 29 44 20)(8 16 53 30 45 21)(9 37 54)(11 25 34)(14 19 28)(17 22 31)

(1 29 4 32 7 35)(2 8 5)(3 31 6 34 9 28)(10 16 13)(11 37 14 40 17 43)(12 38 15 41 18 44)(19 48 22 51 25 54)(20 49 23 52 26 46)(21 27 24)(30 36 33)(39 45 42)(47 53 50)

(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

(1 32)(2 31)(3 30)(4 29)(5 28)(6 36)(7 35)(8 34)(9 33)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 45)(26 44)(27 43)

G:=sub<Sym(54)| (1,18,46,32,38,23)(2,10,47,33,39,24)(3,40,48)(4,12,49,35,41,26)(5,13,50,36,42,27)(6,43,51)(7,15,52,29,44,20)(8,16,53,30,45,21)(9,37,54)(11,25,34)(14,19,28)(17,22,31), (1,29,4,32,7,35)(2,8,5)(3,31,6,34,9,28)(10,16,13)(11,37,14,40,17,43)(12,38,15,41,18,44)(19,48,22,51,25,54)(20,49,23,52,26,46)(21,27,24)(30,36,33)(39,45,42)(47,53,50), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,32)(2,31)(3,30)(4,29)(5,28)(6,36)(7,35)(8,34)(9,33)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,45)(26,44)(27,43)>;

G:=Group( (1,18,46,32,38,23)(2,10,47,33,39,24)(3,40,48)(4,12,49,35,41,26)(5,13,50,36,42,27)(6,43,51)(7,15,52,29,44,20)(8,16,53,30,45,21)(9,37,54)(11,25,34)(14,19,28)(17,22,31), (1,29,4,32,7,35)(2,8,5)(3,31,6,34,9,28)(10,16,13)(11,37,14,40,17,43)(12,38,15,41,18,44)(19,48,22,51,25,54)(20,49,23,52,26,46)(21,27,24)(30,36,33)(39,45,42)(47,53,50), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,32)(2,31)(3,30)(4,29)(5,28)(6,36)(7,35)(8,34)(9,33)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,45)(26,44)(27,43) );

G=PermutationGroup([[(1,18,46,32,38,23),(2,10,47,33,39,24),(3,40,48),(4,12,49,35,41,26),(5,13,50,36,42,27),(6,43,51),(7,15,52,29,44,20),(8,16,53,30,45,21),(9,37,54),(11,25,34),(14,19,28),(17,22,31)], [(1,29,4,32,7,35),(2,8,5),(3,31,6,34,9,28),(10,16,13),(11,37,14,40,17,43),(12,38,15,41,18,44),(19,48,22,51,25,54),(20,49,23,52,26,46),(21,27,24),(30,36,33),(39,45,42),(47,53,50)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,36),(7,35),(8,34),(9,33),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,45),(26,44),(27,43)]])

C₃².₃S₄ is a maximal subgroup of S₃×C₃.S₄
C₃².₃S₄ is a maximal quotient of C₃².₃CSU₂(𝔽₃) C₃².₃GL₂(𝔽₃) C₆².₁₀Dic₃

Matrix representation of C₃².₃S₄ ►in GL₇(𝔽₃₇)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	34	0	0	0
0	0	1	35	0	0	0
0	0	0	0	36	0	0
0	0	0	0	0	36	0
0	0	0	0	0	0	1

0	36	0	0	0	0	0
1	36	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	36	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	36

26	31	0	0	0	0	0
6	20	0	0	0	0	0
0	0	1	34	0	0	0
0	0	1	35	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

0	1	0	0	0	0	0
1	0	0	0	0	0	0
0	0	2	34	0	0	0
0	0	1	35	0	0	0
0	0	0	0	36	0	0
0	0	0	0	0	0	36
0	0	0	0	0	36	0

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,34,35,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[26,6,0,0,0,0,0,31,20,0,0,0,0,0,0,0,1,1,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,34,35,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,36,0] >;

C₃².₃S₄ in GAP, Magma, Sage, TeX

C_3^2._3S_4

% in TeX

G:=Group("C3^2.3S4");

// GroupNames label

G:=SmallGroup(216,94);

// by ID

G=gap.SmallGroup(216,94);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,265,223,218,867,3244,1630,1949,2927]);

// Polycyclic

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

// generators/relations

Export

Character table of C₃².₃S₄ in TeX

G = C32.3S4 order 216 = 23·33

2nd non-split extension by C32 of S4 acting via S4/A4=C2

G = C₃².₃S₄ order 216 = 2³·3³

2^nd non-split extension by C₃² of S₄ acting via S₄/A₄=C₂