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## G = C3×Q8⋊2Dic3order 288 = 25·32

### Direct product of C3 and Q8⋊2Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8⋊2Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4⋊Dic3 — C3×Q8⋊2Dic3
 Lower central C3 — C6 — C12 — C3×Q8⋊2Dic3
 Upper central C1 — C2×C6 — C2×C12 — C6×Q8

Generators and relations for C3×Q82Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >

Subgroups: 202 in 103 conjugacy classes, 50 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 [×2], Q8, C32, Dic3, C12 [×4], C12 [×11], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8, C24, C2×Dic3, C2×C12 [×2], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], Q8⋊C4, C3×Dic3, C3×C12 [×2], C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C6×Q8 [×2], C6×Q8, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, Q82Dic3, C3×Q8⋊C4, C6×C3⋊C8, C3×C4⋊Dic3, Q8×C3×C6, C3×Q82Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, SD16, Q16, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], Q8⋊C4, C3×Dic3 [×2], S3×C6, Q82S3, C3⋊Q16, C6.D4, C3×C22⋊C4, C3×SD16, C3×Q16, C6×Dic3, C3×C3⋊D4 [×2], Q82Dic3, C3×Q8⋊C4, C3×Q82S3, C3×C3⋊Q16, C3×C6.D4, C3×Q82Dic3

Smallest permutation representation of C3×Q82Dic3
On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(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 29 19 12)(2 30 20 7)(3 25 21 8)(4 26 22 9)(5 27 23 10)(6 28 24 11)(13 95 79 87)(14 96 80 88)(15 91 81 89)(16 92 82 90)(17 93 83 85)(18 94 84 86)(31 44 49 39)(32 45 50 40)(33 46 51 41)(34 47 52 42)(35 48 53 37)(36 43 54 38)(55 71 76 63)(56 72 77 64)(57 67 78 65)(58 68 73 66)(59 69 74 61)(60 70 75 62)
(1 46 19 41)(2 47 20 42)(3 48 21 37)(4 43 22 38)(5 44 23 39)(6 45 24 40)(7 52 30 34)(8 53 25 35)(9 54 26 36)(10 49 27 31)(11 50 28 32)(12 51 29 33)(13 68 79 66)(14 69 80 61)(15 70 81 62)(16 71 82 63)(17 72 83 64)(18 67 84 65)(55 90 76 92)(56 85 77 93)(57 86 78 94)(58 87 73 95)(59 88 74 96)(60 89 75 91)
(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 66 4 63)(2 65 5 62)(3 64 6 61)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 81 34 84)(32 80 35 83)(33 79 36 82)(37 85 40 88)(38 90 41 87)(39 89 42 86)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

72 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 12B 12C 12D 12E ··· 12Z 12AA 12AB 12AC 12AD 24A ··· 24H 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 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 2 2 4 4 12 12 1 ··· 1 2 ··· 2 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

72 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 4 4 4 4 type + + + + + + + + - - + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D4 D6 Dic3 SD16 Q16 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 C3×Dic3 C3×SD16 C3×Q16 C3×C3⋊D4 C3×C3⋊D4 Q8⋊2S3 C3⋊Q16 C3×Q8⋊2S3 C3×C3⋊Q16 kernel C3×Q8⋊2Dic3 C6×C3⋊C8 C3×C4⋊Dic3 Q8×C3×C6 Q8⋊2Dic3 Q8×C32 C2×C3⋊C8 C4⋊Dic3 C6×Q8 C3×Q8 C6×Q8 C3×C12 C62 C2×C12 C3×Q8 C3×C6 C3×C6 C2×Q8 C12 C12 C2×C6 C2×C6 C2×C4 Q8 C6 C6 C4 C22 C6 C6 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 1 1 2 2

Matrix representation of C3×Q82Dic3 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 0 72 0 0 0 0 60 7 0 0 7 13
,
 65 0 0 0 30 9 0 0 0 0 1 0 0 0 0 1
,
 21 63 0 0 15 52 0 0 0 0 36 26 0 0 26 37
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,60,7,0,0,7,13],[65,30,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[21,15,0,0,63,52,0,0,0,0,36,26,0,0,26,37] >;

C3×Q82Dic3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("C3xQ8:2Dic3");
// GroupNames label

G:=SmallGroup(288,269);
// by ID

G=gap.SmallGroup(288,269);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,344,2524,1271,102,9414]);
// Polycyclic

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

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