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## G = C3×C2.Dic12order 288 = 25·32

### Direct product of C3 and C2.Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C2.Dic12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C3×C4⋊Dic3 — C3×C2.Dic12
 Lower central C3 — C6 — C12 — C3×C2.Dic12
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C2.Dic12
G = < a,b,c,d | a3=b6=c8=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c3 >

Subgroups: 218 in 95 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×3], C12 [×4], C12 [×5], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C24 [×4], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×3], Q8⋊C4, C3×Dic3 [×3], C3×C12 [×2], C62, C4⋊Dic3, C3×C4⋊C4, C2×C24 [×2], C2×C24, C2×Dic6, C6×Q8, C3×C24, C3×Dic6 [×2], C3×Dic6, C6×Dic3 [×2], C6×C12, C2.Dic12, C3×Q8⋊C4, C3×C4⋊Dic3, C6×C24, C6×Dic6, C3×C2.Dic12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, SD16, Q16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], Q8⋊C4, S3×C6, C24⋊C2, Dic12, D6⋊C4, C3×C22⋊C4, C3×SD16, C3×Q16, S3×C12, C3×D12, C3×C3⋊D4, C2.Dic12, C3×Q8⋊C4, C3×C24⋊C2, C3×Dic12, C3×D6⋊C4, C3×C2.Dic12

Smallest permutation representation of C3×C2.Dic12
On 96 points
Generators in S96
(1 9 65)(2 10 66)(3 11 67)(4 12 68)(5 13 69)(6 14 70)(7 15 71)(8 16 72)(17 29 96)(18 30 89)(19 31 90)(20 32 91)(21 25 92)(22 26 93)(23 27 94)(24 28 95)(33 85 51)(34 86 52)(35 87 53)(36 88 54)(37 81 55)(38 82 56)(39 83 49)(40 84 50)(41 60 73)(42 61 74)(43 62 75)(44 63 76)(45 64 77)(46 57 78)(47 58 79)(48 59 80)
(1 35 9 87 65 53)(2 36 10 88 66 54)(3 37 11 81 67 55)(4 38 12 82 68 56)(5 39 13 83 69 49)(6 40 14 84 70 50)(7 33 15 85 71 51)(8 34 16 86 72 52)(17 73 96 60 29 41)(18 74 89 61 30 42)(19 75 90 62 31 43)(20 76 91 63 32 44)(21 77 92 64 25 45)(22 78 93 57 26 46)(23 79 94 58 27 47)(24 80 95 59 28 48)
(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 62 87 19)(2 22 88 57)(3 60 81 17)(4 20 82 63)(5 58 83 23)(6 18 84 61)(7 64 85 21)(8 24 86 59)(9 75 53 31)(10 26 54 78)(11 73 55 29)(12 32 56 76)(13 79 49 27)(14 30 50 74)(15 77 51 25)(16 28 52 80)(33 92 71 45)(34 48 72 95)(35 90 65 43)(36 46 66 93)(37 96 67 41)(38 44 68 91)(39 94 69 47)(40 42 70 89)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12X 24A ··· 24AF order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 2 12 12 12 12 1 ··· 1 2 ··· 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D4 D6 SD16 Q16 C3×S3 C4×S3 C3⋊D4 C3×D4 D12 C3×D4 S3×C6 C24⋊C2 Dic12 C3×SD16 C3×Q16 S3×C12 C3×C3⋊D4 C3×D12 C3×C24⋊C2 C3×Dic12 kernel C3×C2.Dic12 C3×C4⋊Dic3 C6×C24 C6×Dic6 C2.Dic12 C3×Dic6 C4⋊Dic3 C2×C24 C2×Dic6 Dic6 C2×C24 C3×C12 C62 C2×C12 C3×C6 C3×C6 C2×C8 C12 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8

Matrix representation of C3×C2.Dic12 in GL3(𝔽73) generated by

 8 0 0 0 8 0 0 0 8
,
 72 0 0 0 64 0 0 9 8
,
 27 0 0 0 63 0 0 47 22
,
 46 0 0 0 72 63 0 0 1
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[72,0,0,0,64,9,0,0,8],[27,0,0,0,63,47,0,0,22],[46,0,0,0,72,0,0,63,1] >;

C3×C2.Dic12 in GAP, Magma, Sage, TeX

C_3\times C_2.{\rm Dic}_{12}
% in TeX

G:=Group("C3xC2.Dic12");
// GroupNames label

G:=SmallGroup(288,250);
// by ID

G=gap.SmallGroup(288,250);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,197,260,1683,136,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^8=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^3>;
// generators/relations

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