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## G = S3×C22×C12order 288 = 25·32

### Direct product of C22×C12 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C22×C12
G = < a,b,c,d,e | a2=b2=c12=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 890 in 499 conjugacy classes, 290 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×2], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], S3 [×8], C6 [×2], C6 [×12], C6 [×15], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×28], C2×C6 [×14], C2×C6 [×35], C22×C4, C22×C4 [×13], C24, C3×S3 [×8], C3×C6, C3×C6 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×12], C2×C12 [×28], C22×S3 [×14], C22×C6 [×2], C22×C6 [×15], C23×C4, C3×Dic3 [×4], C3×C12 [×4], S3×C6 [×28], C62 [×7], S3×C2×C4 [×12], C22×Dic3, C22×C12 [×2], C22×C12 [×14], S3×C23, C23×C6, S3×C12 [×16], C6×Dic3 [×6], C6×C12 [×6], S3×C2×C6 [×14], C2×C62, S3×C22×C4, C23×C12, S3×C2×C12 [×12], Dic3×C2×C6, C2×C6×C12, S3×C22×C6, S3×C22×C12
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], S3, C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], D6 [×7], C2×C6 [×35], C22×C4 [×14], C24, C3×S3, C4×S3 [×4], C2×C12 [×28], C22×S3 [×7], C22×C6 [×15], C23×C4, S3×C6 [×7], S3×C2×C4 [×6], C22×C12 [×14], S3×C23, C23×C6, S3×C12 [×4], S3×C2×C6 [×7], S3×C22×C4, C23×C12, S3×C2×C12 [×6], S3×C22×C6, S3×C22×C12

Smallest permutation representation of S3×C22×C12
On 96 points
Generators in S96
(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 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(49 82)(50 83)(51 84)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)

G:=sub<Sym(96)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(49,82)(50,83)(51,84)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87)>;

G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(49,82)(50,83)(51,84)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,85)(47,86)(48,87) );

G=PermutationGroup([(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(49,82),(50,83),(51,84),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,85),(47,86),(48,87)])

144 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 4A ··· 4H 4I ··· 4P 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY 12A ··· 12P 12Q ··· 12AN 12AO ··· 12BD order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 ··· 1 3 ··· 3 1 1 2 2 2 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D6 D6 C3×S3 C4×S3 S3×C6 S3×C6 S3×C12 kernel S3×C22×C12 S3×C2×C12 Dic3×C2×C6 C2×C6×C12 S3×C22×C6 S3×C22×C4 S3×C2×C6 S3×C2×C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 1 1 2 16 24 2 2 2 32 1 6 1 2 8 12 2 16

Matrix representation of S3×C22×C12 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 11 0 0 0 0 4 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 3 0 0 0 1 9
,
 1 0 0 0 0 1 0 0 0 0 10 8 0 0 12 3
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,4,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,3,1,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,10,12,0,0,8,3] >;

S3×C22×C12 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{12}
% in TeX

G:=Group("S3xC2^2xC12");
// GroupNames label

G:=SmallGroup(288,989);
// by ID

G=gap.SmallGroup(288,989);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,192,9414]);
// Polycyclic

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

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