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## G = C2×Dic32order 288 = 25·32

### Direct product of C2, Dic3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×Dic32
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C2×Dic32
 Lower central C32 — C2×Dic32
 Upper central C1 — C23

Generators and relations for C2×Dic32
G = < a,b,c,d,e | a2=b6=d6=1, c2=b3, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 594 in 243 conjugacy classes, 124 normal (10 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×12], C22, C22 [×6], C6 [×14], C6 [×7], C2×C4 [×18], C23, C32, Dic3 [×8], Dic3 [×12], C12 [×8], C2×C6 [×14], C2×C6 [×7], C42 [×4], C22×C4 [×3], C3×C6, C3×C6 [×6], C2×Dic3 [×12], C2×Dic3 [×18], C2×C12 [×12], C22×C6 [×2], C22×C6, C2×C42, C3×Dic3 [×8], C3⋊Dic3 [×4], C62, C62 [×6], C4×Dic3 [×8], C22×Dic3 [×2], C22×Dic3 [×3], C22×C12 [×2], C6×Dic3 [×12], C2×C3⋊Dic3 [×6], C2×C62, C2×C4×Dic3 [×2], Dic32 [×4], Dic3×C2×C6 [×2], C22×C3⋊Dic3, C2×Dic32
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×2], C2×C4 [×18], C23, Dic3 [×8], D6 [×6], C42 [×4], C22×C4 [×3], C4×S3 [×8], C2×Dic3 [×12], C22×S3 [×2], C2×C42, S32, C4×Dic3 [×8], S3×C2×C4 [×4], C22×Dic3 [×2], S3×Dic3 [×4], C6.D6 [×2], C2×S32, C2×C4×Dic3 [×2], Dic32 [×4], C2×S3×Dic3 [×2], C2×C6.D6, C2×Dic32

Smallest permutation representation of C2×Dic32
On 96 points
Generators in S96
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 84)(8 79)(9 80)(10 81)(11 82)(12 83)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(31 52)(32 53)(33 54)(34 49)(35 50)(36 51)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 68)(62 69)(63 70)(64 71)(65 72)(66 67)(85 93)(86 94)(87 95)(88 96)(89 91)(90 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 37 4 40)(2 42 5 39)(3 41 6 38)(7 72 10 69)(8 71 11 68)(9 70 12 67)(13 31 16 34)(14 36 17 33)(15 35 18 32)(19 53 22 50)(20 52 23 49)(21 51 24 54)(25 44 28 47)(26 43 29 46)(27 48 30 45)(55 90 58 87)(56 89 59 86)(57 88 60 85)(61 79 64 82)(62 84 65 81)(63 83 66 80)(73 92 76 95)(74 91 77 94)(75 96 78 93)
(1 13 5 17 3 15)(2 14 6 18 4 16)(7 92 11 96 9 94)(8 93 12 91 10 95)(19 27 23 25 21 29)(20 28 24 26 22 30)(31 39 33 41 35 37)(32 40 34 42 36 38)(43 50 45 52 47 54)(44 51 46 53 48 49)(55 64 57 66 59 62)(56 65 58 61 60 63)(67 77 69 73 71 75)(68 78 70 74 72 76)(79 85 83 89 81 87)(80 86 84 90 82 88)
(1 65 17 60)(2 66 18 55)(3 61 13 56)(4 62 14 57)(5 63 15 58)(6 64 16 59)(7 51 96 48)(8 52 91 43)(9 53 92 44)(10 54 93 45)(11 49 94 46)(12 50 95 47)(19 73 25 67)(20 74 26 68)(21 75 27 69)(22 76 28 70)(23 77 29 71)(24 78 30 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 4A ··· 4P 4Q ··· 4X 6A ··· 6N 6O ··· 6U 12A ··· 12P order 1 2 ··· 2 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 4 3 ··· 3 9 ··· 9 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + - + + + - + + image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 D6 C4×S3 S32 S3×Dic3 C6.D6 C2×S32 kernel C2×Dic32 Dic32 Dic3×C2×C6 C22×C3⋊Dic3 C6×Dic3 C2×C3⋊Dic3 C22×Dic3 C2×Dic3 C2×Dic3 C22×C6 C2×C6 C23 C22 C22 C22 # reps 1 4 2 1 16 8 2 8 4 2 16 1 4 2 1

Matrix representation of C2×Dic32 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 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 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 12 12 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 5 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
,
 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C2×Dic32 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic3^2");
// GroupNames label

G:=SmallGroup(288,602);
// by ID

G=gap.SmallGroup(288,602);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,1356,9414]);
// Polycyclic

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

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