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## G = Dic3×C22×C6order 288 = 25·32

### Direct product of C22×C6 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C22×C6
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C6×Dic3 — Dic3×C2×C6 — Dic3×C22×C6
 Lower central C3 — Dic3×C22×C6
 Upper central C1 — C23×C6

Generators and relations for Dic3×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 810 in 539 conjugacy classes, 370 normal (14 characteristic)
C1, C2, C2 [×14], C3 [×2], C3, C4 [×8], C22 [×35], C6 [×2], C6 [×28], C6 [×15], C2×C4 [×28], C23 [×15], C32, Dic3 [×8], C12 [×8], C2×C6 [×70], C2×C6 [×35], C22×C4 [×14], C24, C3×C6, C3×C6 [×14], C2×Dic3 [×28], C2×C12 [×28], C22×C6 [×30], C22×C6 [×15], C23×C4, C3×Dic3 [×8], C62 [×35], C22×Dic3 [×14], C22×C12 [×14], C23×C6 [×2], C23×C6, C6×Dic3 [×28], C2×C62 [×15], C23×Dic3, C23×C12, Dic3×C2×C6 [×14], C22×C62, Dic3×C22×C6
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], S3, C6 [×15], C2×C4 [×28], C23 [×15], Dic3 [×8], C12 [×8], D6 [×7], C2×C6 [×35], C22×C4 [×14], C24, C3×S3, C2×Dic3 [×28], C2×C12 [×28], C22×S3 [×7], C22×C6 [×15], C23×C4, C3×Dic3 [×8], S3×C6 [×7], C22×Dic3 [×14], C22×C12 [×14], S3×C23, C23×C6, C6×Dic3 [×28], S3×C2×C6 [×7], C23×Dic3, C23×C12, Dic3×C2×C6 [×14], S3×C22×C6, Dic3×C22×C6

Smallest permutation representation of Dic3×C22×C6
On 96 points
Generators in S96
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 71)(8 72)(9 67)(10 68)(11 69)(12 70)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(19 53)(20 54)(21 49)(22 50)(23 51)(24 52)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)(55 89)(56 90)(57 85)(58 86)(59 87)(60 88)(61 81)(62 82)(63 83)(64 84)(65 79)(66 80)(73 93)(74 94)(75 95)(76 96)(77 91)(78 92)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 52)(38 53)(39 54)(40 49)(41 50)(42 51)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(61 76)(62 77)(63 78)(64 73)(65 74)(66 75)(79 94)(80 95)(81 96)(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 14 3 16 5 18)(2 15 4 17 6 13)(7 91 11 95 9 93)(8 92 12 96 10 94)(19 27 21 29 23 25)(20 28 22 30 24 26)(31 39 33 41 35 37)(32 40 34 42 36 38)(43 51 45 53 47 49)(44 52 46 54 48 50)(55 61 59 65 57 63)(56 62 60 66 58 64)(67 73 71 77 69 75)(68 74 72 78 70 76)(79 85 83 89 81 87)(80 86 84 90 82 88)
(1 61 16 57)(2 62 17 58)(3 63 18 59)(4 64 13 60)(5 65 14 55)(6 66 15 56)(7 54 95 44)(8 49 96 45)(9 50 91 46)(10 51 92 47)(11 52 93 48)(12 53 94 43)(19 74 29 70)(20 75 30 71)(21 76 25 72)(22 77 26 67)(23 78 27 68)(24 73 28 69)(31 86 41 82)(32 87 42 83)(33 88 37 84)(34 89 38 79)(35 90 39 80)(36 85 40 81)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2O 3A 3B 3C 3D 3E 4A ··· 4P 6A ··· 6AD 6AE ··· 6BW 12A ··· 12AF order 1 2 ··· 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 2 2 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel Dic3×C22×C6 Dic3×C2×C6 C22×C62 C23×Dic3 C2×C62 C22×Dic3 C23×C6 C22×C6 C23×C6 C22×C6 C22×C6 C24 C23 C23 # reps 1 14 1 2 16 28 2 32 1 8 7 2 16 14

Matrix representation of Dic3×C22×C6 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 3 0 0 0 0 0 4 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 9 0 0 0 0 0 3
,
 12 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,0,4,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,3],[12,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,1,0] >;

Dic3×C22×C6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2^2\times C_6
% in TeX

G:=Group("Dic3xC2^2xC6");
// GroupNames label

G:=SmallGroup(288,1001);
// by ID

G=gap.SmallGroup(288,1001);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,9414]);
// Polycyclic

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

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