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## G = C2×C12⋊S3order 144 = 24·32

### Direct product of C2 and C12⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C12⋊S3
 Lower central C32 — C3×C6 — C2×C12⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C12⋊S3
G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 642 in 162 conjugacy classes, 59 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×16], C6 [×12], C2×C4, D4 [×4], C23 [×2], C32, C12 [×8], D6 [×32], C2×C6 [×4], C2×D4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], D12 [×16], C2×C12 [×4], C22×S3 [×8], C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C2×D12 [×4], C12⋊S3 [×4], C6×C12, C22×C3⋊S3 [×2], C2×C12⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C2×C12⋊S3

Smallest permutation representation of C2×C12⋊S3
On 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 71)(14 72)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 49)(36 50)
(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 29 17)(2 30 18)(3 31 19)(4 32 20)(5 33 21)(6 34 22)(7 35 23)(8 36 24)(9 25 13)(10 26 14)(11 27 15)(12 28 16)(37 52 72)(38 53 61)(39 54 62)(40 55 63)(41 56 64)(42 57 65)(43 58 66)(44 59 67)(45 60 68)(46 49 69)(47 50 70)(48 51 71)
(1 40)(2 39)(3 38)(4 37)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 60)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 72)(33 71)(34 70)(35 69)(36 68)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6L 12A ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 18 18 2 2 2 2 2 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D6 D12 kernel C2×C12⋊S3 C12⋊S3 C6×C12 C22×C3⋊S3 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 4 1 2 4 2 8 4 16

Matrix representation of C2×C12⋊S3 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 6 3 0 0 0 10 3 0 0 0 0 0 6 3 0 0 0 10 3
,
 1 0 0 0 0 0 0 1 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 12 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,6,10,0,0,0,3,3,0,0,0,0,0,6,10,0,0,0,3,3],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

C2×C12⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes S_3
% in TeX

G:=Group("C2xC12:S3");
// GroupNames label

G:=SmallGroup(144,170);
// by ID

G=gap.SmallGroup(144,170);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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