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G = C2×C33⋊C2order 108 = 22·33

Direct product of C2 and C33⋊C2

Aliases: C2×C33⋊C2, C328D6, C336C22, C6⋊(C3⋊S3), (C3×C6)⋊4S3, (C32×C6)⋊3C2, C32(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 [×2], C3 [×13], C22, S3 [×26], C6 [×13], C32 [×13], D6 [×13], C3⋊S3 [×26], C3×C6 [×13], C33, C2×C3⋊S3 [×13], C33⋊C2 [×2], C32×C6, C2×C33⋊C2
Quotients: C1, C2 [×3], C22, S3 [×13], D6 [×13], C3⋊S3 [×13], C2×C3⋊S3 [×13], 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   C338(C2×C4)  C337D4  C338D4  C3312D4  C3315D4  C2×S3×C3⋊S3  C325GL2(𝔽3)
C2×C33⋊C2 is a maximal quotient of   C338Q8  C3312D4  C3315D4

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

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