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## G = C3×Dic6⋊C4order 288 = 25·32

### Direct product of C3 and Dic6⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Dic6⋊C4
G = < a,b,c,d | a3=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, cd=dc >

Subgroups: 274 in 153 conjugacy classes, 86 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×9], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×6], Dic3, C12 [×4], C12 [×15], C2×C6 [×2], C2×C6, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, C3×C6 [×3], Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C3×Q8 [×4], C4×Q8, C3×Dic3 [×6], C3×Dic3, C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], C4×C12 [×3], C3×C4⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6, C6×Q8, C3×Dic6 [×4], C6×Dic3 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], Dic6⋊C4, Q8×C12, Dic3×C12, Dic3×C12 [×2], C3×Dic3⋊C4 [×2], C32×C4⋊C4, C6×Dic6, C3×Dic6⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], Q8 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, C3×S3, C4×S3 [×2], C2×C12 [×6], C3×Q8 [×2], C22×S3, C22×C6, C4×Q8, S3×C6 [×3], S3×C2×C4, D42S3, S3×Q8, C22×C12, C6×Q8, C3×C4○D4, S3×C12 [×2], S3×C2×C6, Dic6⋊C4, Q8×C12, S3×C2×C12, C3×D42S3, C3×S3×Q8, C3×Dic6⋊C4

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 6A ··· 6F 6G ··· 6O 12A ··· 12L 12M ··· 12T 12U ··· 12AL 12AM ··· 12AX order 1 2 2 2 3 3 3 3 3 4 ··· 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 2 2 2 ··· 2 3 3 3 3 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6

90 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 4 4 4 4 type + + + + + + - + - - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 Q8 D6 C4○D4 C3×S3 C3×Q8 C4×S3 S3×C6 C3×C4○D4 S3×C12 D4⋊2S3 S3×Q8 C3×D4⋊2S3 C3×S3×Q8 kernel C3×Dic6⋊C4 Dic3×C12 C3×Dic3⋊C4 C32×C4⋊C4 C6×Dic6 Dic6⋊C4 C3×Dic6 C4×Dic3 Dic3⋊C4 C3×C4⋊C4 C2×Dic6 Dic6 C3×C4⋊C4 C3×Dic3 C2×C12 C3×C6 C4⋊C4 Dic3 C12 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 3 2 1 1 2 8 6 4 2 2 16 1 2 3 2 2 4 4 6 4 8 1 1 2 2

Matrix representation of C3×Dic6⋊C4 in GL5(𝔽13)

 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 9 6 0 0 0 0 3 0 0 0 0 0 8 0 0 0 0 0 5
,
 12 0 0 0 0 0 8 3 0 0 0 5 5 0 0 0 0 0 0 5 0 0 0 5 0
,
 8 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,6,3,0,0,0,0,0,8,0,0,0,0,0,5],[12,0,0,0,0,0,8,5,0,0,0,3,5,0,0,0,0,0,0,5,0,0,0,5,0],[8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×Dic6⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic6:C4");
// GroupNames label

G:=SmallGroup(288,658);
// by ID

G=gap.SmallGroup(288,658);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,303,394,9414]);
// Polycyclic

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

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