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## G = C3×Q8×Dic3order 288 = 25·32

### Direct product of C3, Q8 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Q8×Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 266 in 159 conjugacy classes, 102 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3×C12, C62, C4×Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C6×Q8, C6×Q8, C6×Dic3, C6×Dic3, C6×C12, Q8×C32, Q8×Dic3, Q8×C12, Dic3×C12, C3×C4⋊Dic3, Q8×C3×C6, C3×Q8×Dic3
Quotients:

Smallest permutation representation of C3×Q8×Dic3
On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(1 25 16 21)(2 26 17 22)(3 27 18 23)(4 28 13 24)(5 29 14 19)(6 30 15 20)(7 84 93 89)(8 79 94 90)(9 80 95 85)(10 81 96 86)(11 82 91 87)(12 83 92 88)(31 46 41 49)(32 47 42 50)(33 48 37 51)(34 43 38 52)(35 44 39 53)(36 45 40 54)(55 75 66 70)(56 76 61 71)(57 77 62 72)(58 78 63 67)(59 73 64 68)(60 74 65 69)
(1 37 16 33)(2 38 17 34)(3 39 18 35)(4 40 13 36)(5 41 14 31)(6 42 15 32)(7 77 93 72)(8 78 94 67)(9 73 95 68)(10 74 96 69)(11 75 91 70)(12 76 92 71)(19 49 29 46)(20 50 30 47)(21 51 25 48)(22 52 26 43)(23 53 27 44)(24 54 28 45)(55 87 66 82)(56 88 61 83)(57 89 62 84)(58 90 63 79)(59 85 64 80)(60 86 65 81)
(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 57 4 60)(2 56 5 59)(3 55 6 58)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(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 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

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

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

 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 8
,
 4 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 12 0 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,8],[4,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,12,0,0,0,0,12] >;

C3×Q8×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xQ8xDic3");
// GroupNames label

G:=SmallGroup(288,716);
// by ID

G=gap.SmallGroup(288,716);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,555,268,9414]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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