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## G = C3×C4.Dic6order 288 = 25·32

### Direct product of C3 and C4.Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C4.Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×C4.Dic6
 Lower central C3 — C2×C6 — C3×C4.Dic6
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×C4.Dic6
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 242 in 125 conjugacy classes, 66 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×4], C12 [×4], C12 [×12], C2×C6 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C42.C2, C3×Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×6], C6×Dic3 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], C4.Dic6, C3×C42.C2, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C4⋊Dic3 [×2], C32×C4⋊C4, C3×C4.Dic6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C4○D4 [×2], C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, C42.C2, S3×C6 [×3], C2×Dic6, D42S3, Q83S3, C6×Q8, C3×C4○D4 [×2], C3×Dic6 [×2], S3×C2×C6, C4.Dic6, C3×C42.C2, C6×Dic6, C3×D42S3, C3×Q83S3, C3×C4.Dic6

Smallest permutation representation of C3×C4.Dic6
On 96 points
Generators in S96
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 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 35 70 19)(2 30 71 14)(3 25 72 21)(4 32 61 16)(5 27 62 23)(6 34 63 18)(7 29 64 13)(8 36 65 20)(9 31 66 15)(10 26 67 22)(11 33 68 17)(12 28 69 24)(37 78 52 85)(38 73 53 92)(39 80 54 87)(40 75 55 94)(41 82 56 89)(42 77 57 96)(43 84 58 91)(44 79 59 86)(45 74 60 93)(46 81 49 88)(47 76 50 95)(48 83 51 90)
(1 80 70 87)(2 73 71 92)(3 78 72 85)(4 83 61 90)(5 76 62 95)(6 81 63 88)(7 74 64 93)(8 79 65 86)(9 84 66 91)(10 77 67 96)(11 82 68 89)(12 75 69 94)(13 54 29 39)(14 59 30 44)(15 52 31 37)(16 57 32 42)(17 50 33 47)(18 55 34 40)(19 60 35 45)(20 53 36 38)(21 58 25 43)(22 51 26 48)(23 56 27 41)(24 49 28 46)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G ··· 6O 12A 12B 12C 12D 12E ··· 12Z 12AA ··· 12AH 12AI 12AJ 12AK 12AL order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 1 1 2 2 2 2 2 4 4 6 6 6 6 12 12 1 ··· 1 2 ··· 2 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - - + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 Q8 D6 C4○D4 C3×S3 Dic6 C3×Q8 S3×C6 C3×C4○D4 C3×Dic6 D4⋊2S3 Q8⋊3S3 C3×D4⋊2S3 C3×Q8⋊3S3 kernel C3×C4.Dic6 Dic3×C12 C3×Dic3⋊C4 C3×C4⋊Dic3 C32×C4⋊C4 C4.Dic6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 C3×C4⋊C4 C3×C12 C2×C12 C3×C6 C4⋊C4 C12 C12 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 1 2 3 1 2 2 4 6 2 1 2 3 4 2 4 4 6 8 8 1 1 2 2

Matrix representation of C3×C4.Dic6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 0 12 0 0 1 0 0 0 0 0 9 0 0 0 2 3
,
 0 1 0 0 1 0 0 0 0 0 8 0 0 0 1 5
,
 0 8 0 0 5 0 0 0 0 0 8 2 0 0 0 5
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,12,0,0,0,0,0,9,2,0,0,0,3],[0,1,0,0,1,0,0,0,0,0,8,1,0,0,0,5],[0,5,0,0,8,0,0,0,0,0,8,0,0,0,2,5] >;

C3×C4.Dic6 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_6
% in TeX

G:=Group("C3xC4.Dic6");
// GroupNames label

G:=SmallGroup(288,661);
// by ID

G=gap.SmallGroup(288,661);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1598,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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