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## G = C3×Dic3⋊5D4order 288 = 25·32

### Direct product of C3 and Dic3⋊5D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Dic3⋊5D4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C3×Dic3⋊5D4
 Lower central C3 — C6 — C3×Dic3⋊5D4
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×Dic35D4
G = < a,b,c,d,e | a3=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 466 in 201 conjugacy classes, 86 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], C32, Dic3 [×2], Dic3, C12 [×4], C12 [×11], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×9], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×Dic3, C3×C12 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C4×Dic3, D6⋊C4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C22×C12 [×2], C6×D4, S3×C12 [×4], C3×D12 [×4], C6×Dic3 [×2], C6×C12, C6×C12 [×2], S3×C2×C6 [×2], Dic35D4, D4×C12, Dic3×C12, C3×D6⋊C4 [×2], C32×C4⋊C4, S3×C2×C12 [×2], C6×D12, C3×Dic35D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C3×S3, C4×S3 [×2], C2×C12 [×6], C3×D4 [×2], C22×S3, C22×C6, C4×D4, S3×C6 [×3], S3×C2×C4, S3×D4, Q83S3, C22×C12, C6×D4, C3×C4○D4, S3×C12 [×2], S3×C2×C6, Dic35D4, D4×C12, S3×C2×C12, C3×S3×D4, C3×Q83S3, C3×Dic35D4

Smallest permutation representation of C3×Dic35D4
On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(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 71 4 68)(2 70 5 67)(3 69 6 72)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 77 16 74)(14 76 17 73)(15 75 18 78)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(37 93 40 96)(38 92 41 95)(39 91 42 94)(43 85 46 88)(44 90 47 87)(45 89 48 86)(49 79 52 82)(50 84 53 81)(51 83 54 80)
(1 41 17 34)(2 42 18 35)(3 37 13 36)(4 38 14 31)(5 39 15 32)(6 40 16 33)(7 68 92 76)(8 69 93 77)(9 70 94 78)(10 71 95 73)(11 72 96 74)(12 67 91 75)(19 51 26 43)(20 52 27 44)(21 53 28 45)(22 54 29 46)(23 49 30 47)(24 50 25 48)(55 79 63 87)(56 80 64 88)(57 81 65 89)(58 82 66 90)(59 83 61 85)(60 84 62 86)
(1 88)(2 87)(3 86)(4 85)(5 90)(6 89)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 59)(32 58)(33 57)(34 56)(35 55)(36 60)(37 62)(38 61)(39 66)(40 65)(41 64)(42 63)(43 71)(44 70)(45 69)(46 68)(47 67)(48 72)(49 75)(50 74)(51 73)(52 78)(53 77)(54 76)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 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 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D4 D6 C4○D4 C3×S3 C3×D4 C4×S3 S3×C6 C3×C4○D4 S3×C12 S3×D4 Q8⋊3S3 C3×S3×D4 C3×Q8⋊3S3 kernel C3×Dic3⋊5D4 Dic3×C12 C3×D6⋊C4 C32×C4⋊C4 S3×C2×C12 C6×D12 Dic3⋊5D4 C3×D12 C4×Dic3 D6⋊C4 C3×C4⋊C4 S3×C2×C4 C2×D12 D12 C3×C4⋊C4 C3×Dic3 C2×C12 C3×C6 C4⋊C4 Dic3 C12 C2×C4 C6 C4 C6 C6 C2 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 16 1 2 3 2 2 4 4 6 4 8 1 1 2 2

Matrix representation of C3×Dic35D4 in GL5(𝔽13)

 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 11 0 0 0 1 12
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 11 0 0 0 0 12

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,11,12],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,12] >;

C3×Dic35D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3\rtimes_5D_4
% in TeX

G:=Group("C3xDic3:5D4");
// GroupNames label

G:=SmallGroup(288,664);
// by ID

G=gap.SmallGroup(288,664);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,555,142,9414]);
// Polycyclic

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

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