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## G = C2×C4×C3⋊S3order 144 = 24·32

### Direct product of C2×C4 and C3⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×C3⋊S3
G = < a,b,c,d,e | a2=b4=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 466 in 162 conjugacy classes, 67 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×6], S3 [×16], C6 [×12], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C22×C4, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C4×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, C2×C4×C3⋊S3

Smallest permutation representation of C2×C4×C3⋊S3
On 72 points
Generators in S72
(1 18)(2 19)(3 20)(4 17)(5 68)(6 65)(7 66)(8 67)(9 44)(10 41)(11 42)(12 43)(13 62)(14 63)(15 64)(16 61)(21 47)(22 48)(23 45)(24 46)(25 60)(26 57)(27 58)(28 59)(29 55)(30 56)(31 53)(32 54)(33 50)(34 51)(35 52)(36 49)(37 70)(38 71)(39 72)(40 69)
(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 21 29)(2 22 30)(3 23 31)(4 24 32)(5 16 58)(6 13 59)(7 14 60)(8 15 57)(9 50 69)(10 51 70)(11 52 71)(12 49 72)(17 46 54)(18 47 55)(19 48 56)(20 45 53)(25 66 63)(26 67 64)(27 68 61)(28 65 62)(33 40 44)(34 37 41)(35 38 42)(36 39 43)
(1 33 16)(2 34 13)(3 35 14)(4 36 15)(5 29 44)(6 30 41)(7 31 42)(8 32 43)(9 68 55)(10 65 56)(11 66 53)(12 67 54)(17 49 64)(18 50 61)(19 51 62)(20 52 63)(21 40 58)(22 37 59)(23 38 60)(24 39 57)(25 45 71)(26 46 72)(27 47 69)(28 48 70)
(1 18)(2 19)(3 20)(4 17)(5 69)(6 70)(7 71)(8 72)(9 58)(10 59)(11 60)(12 57)(13 51)(14 52)(15 49)(16 50)(21 55)(22 56)(23 53)(24 54)(25 42)(26 43)(27 44)(28 41)(29 47)(30 48)(31 45)(32 46)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 12A ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 2 2 1 1 1 1 9 9 9 9 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 kernel C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C2×C12 C12 C2×C6 C6 # reps 1 4 1 1 1 8 4 8 4 16

Matrix representation of C2×C4×C3⋊S3 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 5 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 0 1 0 0 12 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 12 0 0 1 0
,
 12 12 0 0 0 1 0 0 0 0 1 0 0 0 12 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[12,0,0,0,12,1,0,0,0,0,1,12,0,0,0,12] >;

C2×C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2xC4xC3:S3");
// GroupNames label

G:=SmallGroup(144,169);
// by ID

G=gap.SmallGroup(144,169);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^4=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,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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