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## G = C4×D6⋊S3order 288 = 25·32

### Direct product of C4 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C4×D6⋊S3
 Lower central C32 — C3×C6 — C4×D6⋊S3
 Upper central C1 — C2×C4

Generators and relations for C4×D6⋊S3
G = < a,b,c,d,e | a4=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 650 in 203 conjugacy classes, 66 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×12], C12 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×7], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C4×Dic3 [×3], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×2], S3×C2×C4 [×2], C2×C3⋊D4 [×2], C22×C12 [×2], D6⋊S3 [×4], S3×C12 [×4], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C4×C3⋊D4 [×2], D6⋊Dic3 [×2], C62.C22, C4×C3⋊Dic3, C2×D6⋊S3, S3×C2×C12 [×2], C4×D6⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], C3⋊D4 [×4], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C4○D12 [×2], C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C4×C3⋊D4 [×2], D6.D6, C4×S32, C2×D6⋊S3, C4×D6⋊S3

Smallest permutation representation of C4×D6⋊S3
On 96 points
Generators in S96
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 67 55 61)(50 68 56 62)(51 69 57 63)(52 70 58 64)(53 71 59 65)(54 72 60 66)(73 91 79 85)(74 92 80 86)(75 93 81 87)(76 94 82 88)(77 95 83 89)(78 96 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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)(49 73)(50 78)(51 77)(52 76)(53 75)(54 74)(55 79)(56 84)(57 83)(58 82)(59 81)(60 80)(61 85)(62 90)(63 89)(64 88)(65 87)(66 86)(67 91)(68 96)(69 95)(70 94)(71 93)(72 92)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(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 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J ··· 6Q 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 ··· 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 6 6 6 6 2 2 4 1 1 1 1 6 6 6 6 18 18 18 18 2 ··· 2 4 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S32 D6⋊S3 C2×S32 D6.D6 C4×S32 kernel C4×D6⋊S3 D6⋊Dic3 C62.C22 C4×C3⋊Dic3 C2×D6⋊S3 S3×C2×C12 D6⋊S3 S3×C2×C4 C3×C12 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 D6 C6 C2×C4 C4 C22 C2 C2 # reps 1 2 1 1 1 2 8 2 2 2 2 2 2 8 8 8 1 2 1 2 2

Matrix representation of C4×D6⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C4×D6⋊S3 in GAP, Magma, Sage, TeX

C_4\times D_6\rtimes S_3
% in TeX

G:=Group("C4xD6:S3");
// GroupNames label

G:=SmallGroup(288,549);
// by ID

G=gap.SmallGroup(288,549);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^6=c^2=d^3=e^2=1,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=b^3*c,e*d*e=d^-1>;
// generators/relations

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