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## G = C2×C3⋊Dic3order 72 = 23·32

### Direct product of C2 and C3⋊Dic3

Aliases: C2×C3⋊Dic3, C6⋊Dic3, C6.15D6, C62.2C2, (C3×C6)⋊3C4, (C2×C6).5S3, C327(C2×C4), C22.(C3⋊S3), C32(C2×Dic3), (C3×C6).14C22, C2.2(C2×C3⋊S3), SmallGroup(72,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3
 Lower central C32 — C2×C3⋊Dic3
 Upper central C1 — C22

Generators and relations for C2×C3⋊Dic3
G = < a,b,c,d | a2=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Character table of C2×C3⋊Dic3

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L size 1 1 1 1 2 2 2 2 9 9 9 9 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 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 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 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 linear of order 2 ρ5 1 -1 1 -1 1 1 1 1 i i -i -i 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 1 1 -i i i -i -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 -i -i i i 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 1 1 i -i -i i -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -1 2 -1 -1 0 0 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ11 2 -2 -2 2 -1 2 -1 -1 0 0 0 0 1 1 1 -1 -2 1 1 1 -1 -1 2 -2 orthogonal lifted from D6 ρ12 2 -2 -2 2 -1 -1 -1 2 0 0 0 0 1 -2 1 -1 1 1 -2 1 2 -1 -1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 1 1 -2 -1 1 1 1 -2 -1 2 -1 1 orthogonal lifted from D6 ρ14 2 2 2 2 -1 -1 -1 2 0 0 0 0 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 2 -1 -1 2 -1 0 0 0 0 2 -1 -1 2 -1 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 -2 -2 2 -1 -1 2 -1 0 0 0 0 -2 1 1 2 1 -2 1 1 -1 -1 -1 1 orthogonal lifted from D6 ρ17 2 -2 2 -2 -1 -1 2 -1 0 0 0 0 2 -1 -1 -2 1 -2 1 1 1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ18 2 2 -2 -2 -1 -1 2 -1 0 0 0 0 -2 1 1 -2 -1 2 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ19 2 2 -2 -2 2 -1 -1 -1 0 0 0 0 1 1 -2 1 -1 -1 -1 2 1 -2 1 1 symplectic lifted from Dic3, Schur index 2 ρ20 2 -2 2 -2 2 -1 -1 -1 0 0 0 0 -1 -1 2 1 1 1 1 -2 1 -2 1 -1 symplectic lifted from Dic3, Schur index 2 ρ21 2 2 -2 -2 -1 -1 -1 2 0 0 0 0 1 -2 1 1 -1 -1 2 -1 -2 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ22 2 -2 2 -2 -1 2 -1 -1 0 0 0 0 -1 -1 -1 1 -2 1 1 1 1 1 -2 2 symplectic lifted from Dic3, Schur index 2 ρ23 2 2 -2 -2 -1 2 -1 -1 0 0 0 0 1 1 1 1 2 -1 -1 -1 1 1 -2 -2 symplectic lifted from Dic3, Schur index 2 ρ24 2 -2 2 -2 -1 -1 -1 2 0 0 0 0 -1 2 -1 1 1 1 -2 1 -2 1 1 -1 symplectic lifted from Dic3, Schur index 2

Smallest permutation representation of C2×C3⋊Dic3
Regular action on 72 points
Generators in S72
(1 45)(2 46)(3 47)(4 48)(5 43)(6 44)(7 63)(8 64)(9 65)(10 66)(11 61)(12 62)(13 57)(14 58)(15 59)(16 60)(17 55)(18 56)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)(49 70)(50 71)(51 72)(52 67)(53 68)(54 69)
(1 23 26)(2 24 27)(3 19 28)(4 20 29)(5 21 30)(6 22 25)(7 16 69)(8 17 70)(9 18 71)(10 13 72)(11 14 67)(12 15 68)(31 46 42)(32 47 37)(33 48 38)(34 43 39)(35 44 40)(36 45 41)(49 64 55)(50 65 56)(51 66 57)(52 61 58)(53 62 59)(54 63 60)
(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)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 49 4 52)(2 54 5 51)(3 53 6 50)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 42 16 39)(14 41 17 38)(15 40 18 37)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 65 28 62)(26 64 29 61)(27 63 30 66)(43 72 46 69)(44 71 47 68)(45 70 48 67)

G:=sub<Sym(72)| (1,45)(2,46)(3,47)(4,48)(5,43)(6,44)(7,63)(8,64)(9,65)(10,66)(11,61)(12,62)(13,57)(14,58)(15,59)(16,60)(17,55)(18,56)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,16,69)(8,17,70)(9,18,71)(10,13,72)(11,14,67)(12,15,68)(31,46,42)(32,47,37)(33,48,38)(34,43,39)(35,44,40)(36,45,41)(49,64,55)(50,65,56)(51,66,57)(52,61,58)(53,62,59)(54,63,60), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,49,4,52)(2,54,5,51)(3,53,6,50)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,42,16,39)(14,41,17,38)(15,40,18,37)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,65,28,62)(26,64,29,61)(27,63,30,66)(43,72,46,69)(44,71,47,68)(45,70,48,67)>;

G:=Group( (1,45)(2,46)(3,47)(4,48)(5,43)(6,44)(7,63)(8,64)(9,65)(10,66)(11,61)(12,62)(13,57)(14,58)(15,59)(16,60)(17,55)(18,56)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,16,69)(8,17,70)(9,18,71)(10,13,72)(11,14,67)(12,15,68)(31,46,42)(32,47,37)(33,48,38)(34,43,39)(35,44,40)(36,45,41)(49,64,55)(50,65,56)(51,66,57)(52,61,58)(53,62,59)(54,63,60), (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)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,49,4,52)(2,54,5,51)(3,53,6,50)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,42,16,39)(14,41,17,38)(15,40,18,37)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,65,28,62)(26,64,29,61)(27,63,30,66)(43,72,46,69)(44,71,47,68)(45,70,48,67) );

G=PermutationGroup([(1,45),(2,46),(3,47),(4,48),(5,43),(6,44),(7,63),(8,64),(9,65),(10,66),(11,61),(12,62),(13,57),(14,58),(15,59),(16,60),(17,55),(18,56),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,35),(26,36),(27,31),(28,32),(29,33),(30,34),(49,70),(50,71),(51,72),(52,67),(53,68),(54,69)], [(1,23,26),(2,24,27),(3,19,28),(4,20,29),(5,21,30),(6,22,25),(7,16,69),(8,17,70),(9,18,71),(10,13,72),(11,14,67),(12,15,68),(31,46,42),(32,47,37),(33,48,38),(34,43,39),(35,44,40),(36,45,41),(49,64,55),(50,65,56),(51,66,57),(52,61,58),(53,62,59),(54,63,60)], [(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),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,49,4,52),(2,54,5,51),(3,53,6,50),(7,34,10,31),(8,33,11,36),(9,32,12,35),(13,42,16,39),(14,41,17,38),(15,40,18,37),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,65,28,62),(26,64,29,61),(27,63,30,66),(43,72,46,69),(44,71,47,68),(45,70,48,67)])

C2×C3⋊Dic3 is a maximal subgroup of
Dic32  D6⋊Dic3  Dic3⋊Dic3  C62.C22  C6.Dic6  C12⋊Dic3  C6.11D12  C625C4  C62.C4  C2×S3×Dic3  D6.4D6  C2×C4×C3⋊S3  C12.D6  C6.GL2(𝔽3)
C2×C3⋊Dic3 is a maximal quotient of
C12.58D6  C12⋊Dic3  C625C4

Matrix representation of C2×C3⋊Dic3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 12 12
,
 12 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 3 10 0 0 0 7 10

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,1,12],[12,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,10,10] >;

C2×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xC3:Dic3");
// GroupNames label

G:=SmallGroup(72,34);
// by ID

G=gap.SmallGroup(72,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,323,1204]);
// Polycyclic

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

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