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## G = C3×Dic3.Q8order 288 = 25·32

### Direct product of C3 and Dic3.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×Dic3.Q8
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12 — C3×Dic3.Q8
 Lower central C3 — C2×C6 — C3×Dic3.Q8
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×Dic3.Q8
G = < a,b,c,d,e | a3=b6=d4=1, c2=b3, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b3c, ce=ec, ede-1=b3d-1 >

Subgroups: 242 in 125 conjugacy classes, 62 normal (58 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×8], C22, C6 [×6], C6 [×3], C2×C4 [×3], C2×C4 [×4], C32, Dic3 [×2], Dic3 [×3], C12 [×15], C2×C6 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×6], C2×C12 [×7], C42.C2, C3×Dic3 [×2], C3×Dic3 [×3], C3×C12 [×3], C62, C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×6], C6×Dic3 [×4], C6×C12 [×3], Dic3.Q8, C3×C42.C2, Dic3×C12, C3×Dic3⋊C4 [×4], C3×C4⋊Dic3, C32×C4⋊C4, C3×Dic3.Q8
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, C3×Q8 [×2], C22×S3, C22×C6, C42.C2, S3×C6 [×3], C4○D12, D42S3, S3×Q8, C6×Q8, C3×C4○D4 [×2], S3×C2×C6, Dic3.Q8, C3×C42.C2, C3×C4○D12, C3×D42S3, C3×S3×Q8, C3×Dic3.Q8

Smallest permutation representation of C3×Dic3.Q8
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 83 4 80)(2 82 5 79)(3 81 6 84)(7 19 10 22)(8 24 11 21)(9 23 12 20)(13 85 16 88)(14 90 17 87)(15 89 18 86)(25 96 28 93)(26 95 29 92)(27 94 30 91)(31 59 34 56)(32 58 35 55)(33 57 36 60)(37 62 40 65)(38 61 41 64)(39 66 42 63)(43 71 46 68)(44 70 47 67)(45 69 48 72)(49 75 52 78)(50 74 53 77)(51 73 54 76)
(1 22 16 26)(2 23 17 27)(3 24 18 28)(4 19 13 29)(5 20 14 30)(6 21 15 25)(7 85 95 80)(8 86 96 81)(9 87 91 82)(10 88 92 83)(11 89 93 84)(12 90 94 79)(31 43 41 54)(32 44 42 49)(33 45 37 50)(34 46 38 51)(35 47 39 52)(36 48 40 53)(55 70 66 75)(56 71 61 76)(57 72 62 77)(58 67 63 78)(59 68 64 73)(60 69 65 74)
(1 56 16 61)(2 55 17 66)(3 60 18 65)(4 59 13 64)(5 58 14 63)(6 57 15 62)(7 54 95 43)(8 53 96 48)(9 52 91 47)(10 51 92 46)(11 50 93 45)(12 49 94 44)(19 76 29 71)(20 75 30 70)(21 74 25 69)(22 73 26 68)(23 78 27 67)(24 77 28 72)(31 88 41 83)(32 87 42 82)(33 86 37 81)(34 85 38 80)(35 90 39 79)(36 89 40 84)

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

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

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

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

Matrix representation of C3×Dic3.Q8 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 8 0 0 8 0 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
,
 0 1 0 0 1 0 0 0 0 0 4 3 0 0 3 9
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,8,0,0,8,0,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,3,0,0,3,9] >;

C3×Dic3.Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3.Q8");
// GroupNames label

G:=SmallGroup(288,660);
// by ID

G=gap.SmallGroup(288,660);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,590,555,268,9414]);
// Polycyclic

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

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