C2xC3^3:C2 - GroupNames

G = C₂×C₃³⋊C₂ order 108 = 2²·3³

Direct product of C₂ and C₃³⋊C₂

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C₂×C₃³⋊C₂, C₃²⋊₈D₆, C₃³⋊₆C₂², C₆⋊(C₃⋊S₃), (C₃×C₆)⋊₄S₃, (C₃²×C₆)⋊₃C₂, C₃⋊₂(C₂×C₃⋊S₃), SmallGroup(108,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×C₃³⋊C₂

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃³⋊C₂ — C₂×C₃³⋊C₂

Lower central

C₃³ — C₂×C₃³⋊C₂

Upper central

C₁ — C₂

Generators and relations for C₂×C₃³⋊C₂
G = < a,b,c,d,e | a²=b³=c³=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b^-1, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 608 in 140 conjugacy classes, 59 normal (5 characteristic)
C₁, C₂, C₂, C₃, C₂², S₃, C₆, C₃², D₆, C₃⋊S₃, C₃×C₆, C₃³, C₂×C₃⋊S₃, C₃³⋊C₂, C₃²×C₆, C₂×C₃³⋊C₂
Quotients: C₁, C₂, C₂², S₃, D₆, C₃⋊S₃, C₂×C₃⋊S₃, C₃³⋊C₂, C₂×C₃³⋊C₂

Character table of C₂×C₃³⋊C₂

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	3L	3M	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M
size	1	1	27	27	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
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	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
ρ₄	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
ρ₅	2	2	0	0	2	2	-1	-1	-1	-1	2	-1	2	-1	-1	-1	-1	-1	2	2	-1	-1	-1	-1	2	-1	2	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	-2	0	0	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	2	-1	1	1	1	-2	1	1	-2	1	1	-2	1	1	-2	orthogonal lifted from D₆
ρ₇	2	-2	0	0	-1	2	-1	-1	2	-1	-1	2	-1	-1	-1	2	-1	1	1	-2	1	1	-2	1	1	-2	1	1	1	-2	orthogonal lifted from D₆
ρ₈	2	2	0	0	-1	2	-1	-1	2	-1	-1	2	-1	-1	-1	2	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	-1	2	orthogonal lifted from S₃
ρ₉	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	2	-1	orthogonal lifted from S₃
ρ₁₀	2	2	0	0	2	-1	2	-1	2	-1	-1	-1	-1	2	-1	-1	-1	-1	2	-1	2	-1	2	-1	-1	-1	-1	2	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	-2	0	0	2	-1	2	-1	2	-1	-1	-1	-1	2	-1	-1	-1	1	-2	1	-2	1	-2	1	1	1	1	-2	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	0	0	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	orthogonal lifted from S₃
ρ₁₃	2	2	0	0	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	2	-1	-1	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	2	orthogonal lifted from S₃
ρ₁₄	2	-2	0	0	-1	2	-1	-1	-1	2	-1	-1	-1	2	-1	-1	2	-2	1	-2	1	1	1	-2	1	1	1	-2	1	1	orthogonal lifted from D₆
ρ₁₅	2	2	0	0	2	-1	-1	2	-1	2	-1	2	-1	-1	-1	-1	-1	-1	2	-1	-1	2	-1	2	-1	2	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₆	2	-2	0	0	-1	2	2	2	-1	-1	-1	-1	-1	-1	2	-1	-1	1	1	-2	-2	-2	1	1	1	1	1	1	-2	1	orthogonal lifted from D₆
ρ₁₇	2	2	0	0	-1	-1	2	-1	-1	-1	2	2	-1	-1	-1	-1	2	2	-1	-1	2	-1	-1	-1	2	2	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₈	2	-2	0	0	2	2	-1	-1	-1	-1	2	-1	2	-1	-1	-1	-1	1	-2	-2	1	1	1	1	-2	1	-2	1	1	1	orthogonal lifted from D₆
ρ₁₉	2	2	0	0	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	2	orthogonal lifted from S₃
ρ₂₀	2	2	0	0	-1	-1	-1	-1	2	2	2	-1	-1	-1	2	-1	-1	-1	-1	-1	-1	-1	2	2	2	-1	-1	-1	2	-1	orthogonal lifted from S₃
ρ₂₁	2	-2	0	0	-1	-1	-1	-1	2	2	2	-1	-1	-1	2	-1	-1	1	1	1	1	1	-2	-2	-2	1	1	1	-2	1	orthogonal lifted from D₆
ρ₂₂	2	-2	0	0	-1	-1	-1	-1	-1	-1	-1	2	2	2	2	-1	-1	1	1	1	1	1	1	1	1	-2	-2	-2	-2	1	orthogonal lifted from D₆
ρ₂₃	2	-2	0	0	-1	-1	-1	2	2	-1	-1	-1	2	-1	-1	-1	2	-2	1	1	1	-2	-2	1	1	1	-2	1	1	1	orthogonal lifted from D₆
ρ₂₄	2	-2	0	0	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	-2	-2	1	1	1	1	1	1	1	1	1	-2	-2	orthogonal lifted from D₆
ρ₂₅	2	-2	0	0	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	2	-1	1	1	1	1	-2	1	1	-2	1	1	-2	1	-2	orthogonal lifted from D₆
ρ₂₆	2	2	0	0	-1	-1	-1	2	2	-1	-1	-1	2	-1	-1	-1	2	2	-1	-1	-1	2	2	-1	-1	-1	2	-1	-1	-1	orthogonal lifted from S₃
ρ₂₇	2	2	0	0	-1	2	-1	-1	-1	2	-1	-1	-1	2	-1	-1	2	2	-1	2	-1	-1	-1	2	-1	-1	-1	2	-1	-1	orthogonal lifted from S₃
ρ₂₈	2	-2	0	0	2	-1	-1	2	-1	2	-1	2	-1	-1	-1	-1	-1	1	-2	1	1	-2	1	-2	1	-2	1	1	1	1	orthogonal lifted from D₆
ρ₂₉	2	-2	0	0	-1	-1	2	-1	-1	-1	2	2	-1	-1	-1	-1	2	-2	1	1	-2	1	1	1	-2	-2	1	1	1	1	orthogonal lifted from D₆
ρ₃₀	2	2	0	0	-1	2	2	2	-1	-1	-1	-1	-1	-1	2	-1	-1	-1	-1	2	2	2	-1	-1	-1	-1	-1	-1	2	-1	orthogonal lifted from S₃

Smallest permutation representation of C₂×C₃³⋊C₂
►On 54 points

Generators in S₅₄

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

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

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

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

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

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

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

C₂×C₃³⋊C₂ is a maximal subgroup of C₃³⋊₈(C₂×C₄) C₃³⋊₇D₄ C₃³⋊₈D₄ C₃³⋊₁₂D₄ C₃³⋊₁₅D₄ C₂×S₃×C₃⋊S₃ C₃²⋊₅GL₂(𝔽₃)
C₂×C₃³⋊C₂ is a maximal quotient of C₃³⋊₈Q₈ C₃³⋊₁₂D₄ C₃³⋊₁₅D₄

Matrix representation of C₂×C₃³⋊C₂ ►in GL₆(ℤ)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	-1	-1	0	0
0	0	1	0	0	0
0	0	0	0	-1	-1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	-1	-1
0	0	0	0	1	0

0	1	0	0	0	0
-1	-1	0	0	0	0
0	0	0	1	0	0
0	0	-1	-1	0	0
0	0	0	0	0	1
0	0	0	0	-1	-1

1	0	0	0	0	0
-1	-1	0	0	0	0
0	0	-1	0	0	0
0	0	1	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂×C₃³⋊C₂ in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes C_2

% in TeX

G:=Group("C2xC3^3:C2");

// GroupNames label

G:=SmallGroup(108,44);

// by ID

G=gap.SmallGroup(108,44);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,122,483,1804]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂×C₃³⋊C₂ in TeX

G = C2×C33⋊C2 order 108 = 22·33

Direct product of C2 and C33⋊C2

G = C₂×C₃³⋊C₂ order 108 = 2²·3³

Direct product of C₂ and C₃³⋊C₂