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## G = C2×D12.S3order 288 = 25·32

### Direct product of C2 and D12.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×D12.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — C2×D12.S3
 Lower central C32 — C3×C6 — C3×C12 — C2×D12.S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D12.S3
G = < a,b,c,d,e | a2=b12=c2=d3=1, e2=b9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 610 in 155 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C32, Dic3 [×8], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×10], D12 [×2], D12, C2×Dic3 [×4], C2×C12 [×2], C2×C12, C3×D4 [×3], C22×S3, C22×C6, C2×SD16, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×4], C62, C24⋊C2 [×4], C2×C3⋊C8, D4.S3 [×4], C2×C24, C2×Dic6 [×3], C2×D12, C6×D4, C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C2×C24⋊C2, C2×D4.S3, D12.S3 [×4], C6×C3⋊C8, C6×D12, C2×C324Q8, C2×D12.S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C2×SD16, S32, C24⋊C2 [×2], D4.S3 [×2], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C2×C24⋊C2, C2×D4.S3, D12.S3 [×2], C2×C3⋊D12, C2×D12.S3

Smallest permutation representation of C2×D12.S3
On 96 points
Generators in S96
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 53)(14 52)(15 51)(16 50)(17 49)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 87)(26 86)(27 85)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(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 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(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 34 10 31 7 28 4 25)(2 35 11 32 8 29 5 26)(3 36 12 33 9 30 6 27)(13 45 22 42 19 39 16 48)(14 46 23 43 20 40 17 37)(15 47 24 44 21 41 18 38)(49 73 58 82 55 79 52 76)(50 74 59 83 56 80 53 77)(51 75 60 84 57 81 54 78)(61 85 70 94 67 91 64 88)(62 86 71 95 68 92 65 89)(63 87 72 96 69 93 66 90)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 12 12 2 2 4 2 2 36 36 2 ··· 2 4 4 4 12 12 12 12 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 SD16 D12 C3⋊D4 D12 C3⋊D4 C24⋊C2 S32 D4.S3 C3⋊D12 C2×S32 C3⋊D12 D12.S3 kernel C2×D12.S3 D12.S3 C6×C3⋊C8 C6×D12 C2×C32⋊4Q8 C2×C3⋊C8 C2×D12 C3×C12 C62 C3⋊C8 D12 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 1 1 1 4

Matrix representation of C2×D12.S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 59 7 0 0 66 66
,
 1 0 0 0 0 1 0 0 0 0 66 66 0 0 59 7
,
 0 72 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 36 62 0 0 11 25
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,59,66,0,0,7,66],[1,0,0,0,0,1,0,0,0,0,66,59,0,0,66,7],[0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,11,0,0,62,25] >;

C2×D12.S3 in GAP, Magma, Sage, TeX

C_2\times D_{12}.S_3
% in TeX

G:=Group("C2xD12.S3");
// GroupNames label

G:=SmallGroup(288,476);
// by ID

G=gap.SmallGroup(288,476);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^3=1,e^2=b^9,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^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations

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