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G = C2×C3⋊Dic3order 72 = 23·32

Direct product of C2 and C3⋊Dic3

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

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

C1C32 — C2×C3⋊Dic3
C1C3C32C3×C6C3⋊Dic3 — C2×C3⋊Dic3
C32 — C2×C3⋊Dic3
C1C22

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 >

9C4
9C4
9C2×C4
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3Dic3
3C2×Dic3
3C2×Dic3
3C2×Dic3
3C2×Dic3

Character table of C2×C3⋊Dic3

 class 12A2B2C3A3B3C3D4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L
 size 111122229999222222222222
ρ1111111111111111111111111    trivial
ρ21-1-111111-11-11-1-1-11-1-1-1-1111-1    linear of order 2
ρ311111111-1-1-1-1111111111111    linear of order 2
ρ41-1-1111111-11-1-1-1-11-1-1-1-1111-1    linear of order 2
ρ51-11-11111ii-i-i111-1-1-1-1-1-1-1-11    linear of order 4
ρ611-1-11111-iii-i-1-1-1-11111-1-1-1-1    linear of order 4
ρ71-11-11111-i-iii111-1-1-1-1-1-1-1-11    linear of order 4
ρ811-1-11111i-i-ii-1-1-1-11111-1-1-1-1    linear of order 4
ρ922222-1-1-10000-1-12-1-1-1-12-12-1-1    orthogonal lifted from S3
ρ102222-12-1-10000-1-1-1-12-1-1-1-1-122    orthogonal lifted from S3
ρ112-2-22-12-1-10000111-1-2111-1-12-2    orthogonal lifted from D6
ρ122-2-22-1-1-1200001-21-111-212-1-11    orthogonal lifted from D6
ρ132-2-222-1-1-1000011-2-1111-2-12-11    orthogonal lifted from D6
ρ142222-1-1-120000-12-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ152222-1-12-100002-1-12-12-1-1-1-1-1-1    orthogonal lifted from S3
ρ162-2-22-1-12-10000-21121-211-1-1-11    orthogonal lifted from D6
ρ172-22-2-1-12-100002-1-1-21-211111-1    symplectic lifted from Dic3, Schur index 2
ρ1822-2-2-1-12-10000-211-2-12-1-11111    symplectic lifted from Dic3, Schur index 2
ρ1922-2-22-1-1-1000011-21-1-1-121-211    symplectic lifted from Dic3, Schur index 2
ρ202-22-22-1-1-10000-1-121111-21-21-1    symplectic lifted from Dic3, Schur index 2
ρ2122-2-2-1-1-1200001-211-1-12-1-2111    symplectic lifted from Dic3, Schur index 2
ρ222-22-2-12-1-10000-1-1-11-211111-22    symplectic lifted from Dic3, Schur index 2
ρ2322-2-2-12-1-1000011112-1-1-111-2-2    symplectic lifted from Dic3, Schur index 2
ρ242-22-2-1-1-120000-12-1111-21-211-1    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)

10000
012000
001200
00010
00001
,
10000
09000
00300
00001
0001212
,
120000
09000
00300
00010
00001
,
50000
00100
01000
000310
000710

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

Export

Subgroup lattice of C2×C3⋊Dic3 in TeX
Character table of C2×C3⋊Dic3 in TeX

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