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## G = Dic3×C2×C12order 288 = 25·32

### Direct product of C2×C12 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C2×C6 — Dic3×C2×C12
 Lower central C3 — Dic3×C2×C12
 Upper central C1 — C22×C12

Generators and relations for Dic3×C2×C12
G = < a,b,c,d | a2=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 378 in 243 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×4], C4 [×8], C22, C22 [×6], C6 [×2], C6 [×12], C6 [×7], C2×C4 [×6], C2×C4 [×12], C23, C32, Dic3 [×8], C12 [×8], C12 [×12], C2×C6 [×2], C2×C6 [×12], C2×C6 [×7], C42 [×4], C22×C4, C22×C4 [×2], C3×C6, C3×C6 [×6], C2×Dic3 [×12], C2×C12 [×12], C2×C12 [×18], C22×C6 [×2], C22×C6, C2×C42, C3×Dic3 [×8], C3×C12 [×4], C62, C62 [×6], C4×Dic3 [×4], C4×C12 [×4], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Dic3 [×12], C6×C12 [×6], C2×C62, C2×C4×Dic3, C2×C4×C12, Dic3×C12 [×4], Dic3×C2×C6 [×2], C2×C6×C12, Dic3×C2×C12
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], S3, C6 [×7], C2×C4 [×18], C23, Dic3 [×4], C12 [×12], D6 [×3], C2×C6 [×7], C42 [×4], C22×C4 [×3], C3×S3, C4×S3 [×4], C2×Dic3 [×6], C2×C12 [×18], C22×S3, C22×C6, C2×C42, C3×Dic3 [×4], S3×C6 [×3], C4×Dic3 [×4], C4×C12 [×4], S3×C2×C4 [×2], C22×Dic3, C22×C12 [×3], S3×C12 [×4], C6×Dic3 [×6], S3×C2×C6, C2×C4×Dic3, C2×C4×C12, Dic3×C12 [×4], S3×C2×C12 [×2], Dic3×C2×C6, Dic3×C2×C12

Smallest permutation representation of Dic3×C2×C12
On 96 points
Generators in S96
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)
(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 67 5 71 9 63)(2 68 6 72 10 64)(3 69 7 61 11 65)(4 70 8 62 12 66)(13 83 21 79 17 75)(14 84 22 80 18 76)(15 73 23 81 19 77)(16 74 24 82 20 78)(25 39 29 43 33 47)(26 40 30 44 34 48)(27 41 31 45 35 37)(28 42 32 46 36 38)(49 88 57 96 53 92)(50 89 58 85 54 93)(51 90 59 86 55 94)(52 91 60 87 56 95)
(1 56 71 91)(2 57 72 92)(3 58 61 93)(4 59 62 94)(5 60 63 95)(6 49 64 96)(7 50 65 85)(8 51 66 86)(9 52 67 87)(10 53 68 88)(11 54 69 89)(12 55 70 90)(13 44 79 26)(14 45 80 27)(15 46 81 28)(16 47 82 29)(17 48 83 30)(18 37 84 31)(19 38 73 32)(20 39 74 33)(21 40 75 34)(22 41 76 35)(23 42 77 36)(24 43 78 25)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A ··· 4H 4I ··· 4X 6A ··· 6N 6O ··· 6AI 12A ··· 12P 12Q ··· 12AN 12AO ··· 12BT order 1 2 ··· 2 3 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 2 2 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 Dic3 D6 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C6 S3×C12 kernel Dic3×C2×C12 Dic3×C12 Dic3×C2×C6 C2×C6×C12 C2×C4×Dic3 C6×Dic3 C6×C12 C4×Dic3 C22×Dic3 C22×C12 C2×Dic3 C2×C12 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 2 16 8 8 4 2 32 16 1 4 2 1 2 8 8 4 2 16

Matrix representation of Dic3×C2×C12 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 10 0 0 0 0 0 10
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 3
,
 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,10],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

Dic3×C2×C12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{12}
% in TeX

G:=Group("Dic3xC2xC12");
// GroupNames label

G:=SmallGroup(288,693);
// by ID

G=gap.SmallGroup(288,693);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,268,9414]);
// Polycyclic

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

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