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## G = C3×Dic24order 288 = 25·32

### Direct product of C3 and Dic24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×Dic24
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×Dic12 — C3×Dic24
 Lower central C3 — C6 — C12 — C24 — C3×Dic24
 Upper central C1 — C6 — C12 — C24 — C48

Generators and relations for C3×Dic24
G = < a,b,c | a3=b48=1, c2=b24, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic24
On 96 points
Generators in S96
(1 33 17)(2 34 18)(3 35 19)(4 36 20)(5 37 21)(6 38 22)(7 39 23)(8 40 24)(9 41 25)(10 42 26)(11 43 27)(12 44 28)(13 45 29)(14 46 30)(15 47 31)(16 48 32)(49 65 81)(50 66 82)(51 67 83)(52 68 84)(53 69 85)(54 70 86)(55 71 87)(56 72 88)(57 73 89)(58 74 90)(59 75 91)(60 76 92)(61 77 93)(62 78 94)(63 79 95)(64 80 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 73 25 49)(2 72 26 96)(3 71 27 95)(4 70 28 94)(5 69 29 93)(6 68 30 92)(7 67 31 91)(8 66 32 90)(9 65 33 89)(10 64 34 88)(11 63 35 87)(12 62 36 86)(13 61 37 85)(14 60 38 84)(15 59 39 83)(16 58 40 82)(17 57 41 81)(18 56 42 80)(19 55 43 79)(20 54 44 78)(21 53 45 77)(22 52 46 76)(23 51 47 75)(24 50 48 74)

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

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

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

81 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A ··· 12H 12I 12J 12K 12L 16A 16B 16C 16D 24A ··· 24P 48A ··· 48AF order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 12 ··· 12 12 12 12 12 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 1 1 2 2 2 2 24 24 1 1 2 2 2 2 2 2 ··· 2 24 24 24 24 2 2 2 2 2 ··· 2 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 D8 C3×S3 D12 C3×D4 Q32 S3×C6 D24 C3×D8 C3×D12 Dic24 C3×Q32 C3×D24 C3×Dic24 kernel C3×Dic24 C3×C48 C3×Dic12 Dic24 C48 Dic12 C48 C3×C12 C24 C3×C6 C16 C12 C12 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 2 4 4 4 8 8 8 16

Matrix representation of C3×Dic24 in GL2(𝔽97) generated by

 35 0 0 35
,
 31 0 0 72
,
 0 1 96 0
G:=sub<GL(2,GF(97))| [35,0,0,35],[31,0,0,72],[0,96,1,0] >;

C3×Dic24 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{24}
% in TeX

G:=Group("C3xDic24");
// GroupNames label

G:=SmallGroup(288,235);
// by ID

G=gap.SmallGroup(288,235);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,197,260,1011,192,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^48=1,c^2=b^24,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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