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## G = C2×D6⋊S3order 144 = 24·32

### Direct product of C2 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — D6⋊S3 — C2×D6⋊S3
 Lower central C32 — C3×C6 — C2×D6⋊S3
 Upper central C1 — C22

Generators and relations for C2×D6⋊S3
G = < a,b,c,d,e | a2=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 352 in 116 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, D4 [×4], C23 [×2], C32, Dic3 [×6], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C2×D4, C3×S3 [×4], C3×C6, C3×C6 [×2], C2×Dic3 [×3], C3⋊D4 [×8], C22×S3 [×2], C22×C6 [×2], C3⋊Dic3 [×2], S3×C6 [×4], S3×C6 [×4], C62, C2×C3⋊D4 [×2], D6⋊S3 [×4], C2×C3⋊Dic3, S3×C2×C6 [×2], C2×D6⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×4], C22×S3 [×2], S32, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C2×D6⋊S3

Character table of C2×D6⋊S3

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q size 1 1 1 1 6 6 6 6 2 2 4 18 18 2 2 2 2 2 2 4 4 4 6 6 6 6 6 6 6 6 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 -2 2 0 0 0 0 2 2 2 0 0 2 2 -2 -2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 0 2 -1 -1 0 0 -1 2 -1 -1 2 2 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 0 0 0 0 2 2 2 0 0 -2 -2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 2 -2 0 2 -2 0 2 -1 -1 0 0 1 -2 -1 1 -2 2 1 -1 1 -1 0 0 -1 0 0 1 1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 0 0 -2 -1 2 -1 0 0 -2 1 2 -2 1 -1 1 -1 1 0 1 1 0 -1 -1 0 0 orthogonal lifted from D6 ρ14 2 -2 2 -2 -2 0 0 2 -1 2 -1 0 0 -2 1 2 -2 1 -1 1 -1 1 0 -1 -1 0 1 1 0 0 orthogonal lifted from D6 ρ15 2 2 2 2 2 0 0 2 -1 2 -1 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ16 2 2 2 2 -2 0 0 -2 -1 2 -1 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ17 2 2 2 2 0 -2 -2 0 2 -1 -1 0 0 -1 2 -1 -1 2 2 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ18 2 -2 2 -2 0 -2 2 0 2 -1 -1 0 0 1 -2 -1 1 -2 2 1 -1 1 1 0 0 1 0 0 -1 -1 orthogonal lifted from D6 ρ19 2 2 -2 -2 0 0 0 0 2 -1 -1 0 0 1 -2 1 -1 2 -2 1 1 -1 -√-3 0 0 √-3 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -1 2 1 1 -2 -2 -1 1 1 √-3 0 0 -√-3 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 0 0 -1 2 -1 0 0 -2 1 -2 2 -1 1 1 1 -1 0 -√-3 √-3 0 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ22 2 -2 -2 2 0 0 0 0 -1 2 -1 0 0 2 -1 -2 -2 1 1 -1 1 1 0 √-3 -√-3 0 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ23 2 -2 -2 2 0 0 0 0 -1 2 -1 0 0 2 -1 -2 -2 1 1 -1 1 1 0 -√-3 √-3 0 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ24 2 2 -2 -2 0 0 0 0 -1 2 -1 0 0 -2 1 -2 2 -1 1 1 1 -1 0 √-3 -√-3 0 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ25 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -1 2 1 1 -2 -2 -1 1 1 -√-3 0 0 √-3 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ26 2 2 -2 -2 0 0 0 0 2 -1 -1 0 0 1 -2 1 -1 2 -2 1 1 -1 √-3 0 0 -√-3 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ27 4 4 4 4 0 0 0 0 -2 -2 1 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ28 4 -4 4 -4 0 0 0 0 -2 -2 1 0 0 2 2 -2 2 2 -2 -1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ29 4 4 -4 -4 0 0 0 0 -2 -2 1 0 0 2 2 2 -2 -2 2 -1 -1 1 0 0 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ30 4 -4 -4 4 0 0 0 0 -2 -2 1 0 0 -2 -2 2 2 2 2 1 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2

Smallest permutation representation of C2×D6⋊S3
On 48 points
Generators in S48
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(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)
(1 22)(2 21)(3 20)(4 19)(5 24)(6 23)(7 16)(8 15)(9 14)(10 13)(11 18)(12 17)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)

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

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

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

Matrix representation of C2×D6⋊S3 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
,
 -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 0 -1 0 0 0 0 1 -1
,
 -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 0 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1],[-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,-1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D6⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes S_3
% in TeX

G:=Group("C2xD6:S3");
// GroupNames label

G:=SmallGroup(144,150);
// by ID

G=gap.SmallGroup(144,150);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,490,3461]);
// Polycyclic

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

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