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## G = C2×Dic6⋊S3order 288 = 25·32

### Direct product of C2 and Dic6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×Dic6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — Dic6⋊S3 — C2×Dic6⋊S3
 Lower central C32 — C3×C6 — C3×C12 — C2×Dic6⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic6⋊S3
G = < a,b,c,d,e | a2=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 498 in 147 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 [×2], C12 [×4], C12 [×4], 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 [×6], Dic6 [×2], Dic6, D12 [×2], D12, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×3], C3×Q8 [×3], C22×S3, C22×C6, C2×SD16, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×4], C62, C2×C3⋊C8 [×3], D4.S3 [×4], Q82S3 [×4], C2×Dic6, C2×D12, C6×D4, C6×Q8, C324C8 [×2], C3×Dic6 [×2], C3×Dic6, C3×D12 [×2], C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2×D4.S3, C2×Q82S3, Dic6⋊S3 [×4], C2×C324C8, C6×Dic6, C6×D12, C2×Dic6⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C2×SD16, S32, D4.S3 [×2], Q82S3 [×2], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C2×D4.S3, C2×Q82S3, Dic6⋊S3 [×2], C2×D6⋊S3, C2×Dic6⋊S3

Smallest permutation representation of C2×Dic6⋊S3
On 96 points
Generators in S96
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(73 93)(74 94)(75 95)(76 96)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(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 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 45 19 39)(14 44 20 38)(15 43 21 37)(16 42 22 48)(17 41 23 47)(18 40 24 46)(49 74 55 80)(50 73 56 79)(51 84 57 78)(52 83 58 77)(53 82 59 76)(54 81 60 75)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)
(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 70)(2 65)(3 72)(4 67)(5 62)(6 69)(7 64)(8 71)(9 66)(10 61)(11 68)(12 63)(13 55)(14 50)(15 57)(16 52)(17 59)(18 54)(19 49)(20 56)(21 51)(22 58)(23 53)(24 60)(25 91)(26 86)(27 93)(28 88)(29 95)(30 90)(31 85)(32 92)(33 87)(34 94)(35 89)(36 96)(37 81)(38 76)(39 83)(40 78)(41 73)(42 80)(43 75)(44 82)(45 77)(46 84)(47 79)(48 74)

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

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

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

42 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 ··· 12H 12I 12J 12K 12L 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 size 1 1 1 1 12 12 2 2 4 2 2 12 12 2 ··· 2 4 4 4 12 12 12 12 18 18 18 18 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 SD16 C3⋊D4 C3⋊D4 S32 D4.S3 Q8⋊2S3 D6⋊S3 C2×S32 D6⋊S3 Dic6⋊S3 kernel C2×Dic6⋊S3 Dic6⋊S3 C2×C32⋊4C8 C6×Dic6 C6×D12 C2×Dic6 C2×D12 C3×C12 C62 Dic6 D12 C2×C12 C3×C6 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 1 2 2 2 4 4 4 1 2 2 1 1 1 4

Matrix representation of C2×Dic6⋊S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 19 21 0 0 0 0 21 54 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 30 13 0 0 0 0 60 43
,
 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 72 0 0 0 0 1 72
,
 52 61 0 0 0 0 61 21 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 50 5 0 0 0 0 55 23

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[19,21,0,0,0,0,21,54,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[52,61,0,0,0,0,61,21,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,50,55,0,0,0,0,5,23] >;

C2×Dic6⋊S3 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_6\rtimes S_3
% in TeX

G:=Group("C2xDic6:S3");
// GroupNames label

G:=SmallGroup(288,474);
// by ID

G=gap.SmallGroup(288,474);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=d^3=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^7,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations

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