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G = C2×C33⋊C2order 108 = 22·33

Direct product of C2 and C33⋊C2

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

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

C1C33 — C2×C33⋊C2
C1C3C32C33C33⋊C2 — C2×C33⋊C2
C33 — C2×C33⋊C2
C1C2

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 12A2B2C3A3B3C3D3E3F3G3H3I3J3K3L3M6A6B6C6D6E6F6G6H6I6J6K6L6M
 size 11272722222222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211-1-111111111111111111111111111    linear of order 2
ρ31-1-111111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41-11-11111111111111-1-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ5220022-1-1-1-12-12-1-1-1-1-122-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ62-200-1-12-1-12-1-12-1-12-1111-211-211-211-2    orthogonal lifted from D6
ρ72-200-12-1-12-1-12-1-1-12-111-211-211-2111-2    orthogonal lifted from D6
ρ82200-12-1-12-1-12-1-1-12-1-1-12-1-12-1-12-1-1-12    orthogonal lifted from S3
ρ92200-1-1-1-1-1-1-12222-1-1-1-1-1-1-1-1-1-12222-1    orthogonal lifted from S3
ρ1022002-12-12-1-1-1-12-1-1-1-12-12-12-1-1-1-12-1-1    orthogonal lifted from S3
ρ112-2002-12-12-1-1-1-12-1-1-11-21-21-21111-211    orthogonal lifted from D6
ρ1222002-1-1-1-1-1-1-1-1-122222-1-1-1-1-1-1-1-1-122    orthogonal lifted from S3
ρ132200-1-1-12-1-12-1-12-12-1-1-1-1-12-1-12-1-12-12    orthogonal lifted from S3
ρ142-200-12-1-1-12-1-1-12-1-12-21-2111-2111-211    orthogonal lifted from D6
ρ1522002-1-12-12-12-1-1-1-1-1-12-1-12-12-12-1-1-1-1    orthogonal lifted from S3
ρ162-200-1222-1-1-1-1-1-12-1-111-2-2-2111111-21    orthogonal lifted from D6
ρ172200-1-12-1-1-122-1-1-1-122-1-12-1-1-122-1-1-1-1    orthogonal lifted from S3
ρ182-20022-1-1-1-12-12-1-1-1-11-2-21111-21-2111    orthogonal lifted from D6
ρ192200-1-12-1-12-1-12-1-12-1-1-1-12-1-12-1-12-1-12    orthogonal lifted from S3
ρ202200-1-1-1-1222-1-1-12-1-1-1-1-1-1-1222-1-1-12-1    orthogonal lifted from S3
ρ212-200-1-1-1-1222-1-1-12-1-111111-2-2-2111-21    orthogonal lifted from D6
ρ222-200-1-1-1-1-1-1-12222-1-111111111-2-2-2-21    orthogonal lifted from D6
ρ232-200-1-1-122-1-1-12-1-1-12-2111-2-2111-2111    orthogonal lifted from D6
ρ242-2002-1-1-1-1-1-1-1-1-1222-2-2111111111-2-2    orthogonal lifted from D6
ρ252-200-1-1-12-1-12-1-12-12-11111-211-211-21-2    orthogonal lifted from D6
ρ262200-1-1-122-1-1-12-1-1-122-1-1-122-1-1-12-1-1-1    orthogonal lifted from S3
ρ272200-12-1-1-12-1-1-12-1-122-12-1-1-12-1-1-12-1-1    orthogonal lifted from S3
ρ282-2002-1-12-12-12-1-1-1-1-11-211-21-21-21111    orthogonal lifted from D6
ρ292-200-1-12-1-1-122-1-1-1-12-211-2111-2-21111    orthogonal lifted from D6
ρ302200-1222-1-1-1-1-1-12-1-1-1-1222-1-1-1-1-1-12-1    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)

100000
010000
00-1000
000-100
000010
000001
,
010000
-1-10000
00-1-100
001000
0000-1-1
000010
,
100000
010000
001000
000100
0000-1-1
000010
,
010000
-1-10000
000100
00-1-100
000001
0000-1-1
,
100000
-1-10000
00-1000
001100
000001
000010

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

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