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G = C3×Dic24  order 288 = 25·32

Direct product of C3 and Dic24

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic24, C48.1C6, C48.5S3, C32⋊4Q32, C6.21D24, C24.81D6, C12.64D12, Dic12.1C6, C16.(C3×S3), C3⋊1(C3×Q32), C6.3(C3×D8), C8.15(S3×C6), (C3×C48).2C2, C2.5(C3×D24), (C3×C6).20D8, C4.3(C3×D12), C24.18(C2×C6), C12.26(C3×D4), (C3×C12).128D4, (C3×C24).56C22, (C3×Dic12).1C2, SmallGroup(288,235)

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 >

C1
C2
C3
C3
2C3
C4
12C4
12C4
C6
C6
2C6
C32
C8
6Q8
6Q8
C12
C12
2C12
4Dic3
4Dic3
12C12
12C12
C3×C6
C16
3Q16
3Q16
C24
C24
2Dic6
2C24
2Dic6
6C3×Q8
6C3×Q8
C3×C12
4C3×Dic3
4C3×Dic3
3Q32
C48
C48
Dic12
Dic12
2C48
3C3×Q16
3C3×Q16
C3×C24
2C3×Dic6
2C3×Dic6
Dic24
3C3×Q32
C3×Dic12
C3×Dic12
C3×C48
C3×Dic24

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

81 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B12A···12H12I12J12K12L16A16B16C16D24A···24P48A···48AF
order1233333444666668812···12121212121616161624···2448···48
size11112222242411222222···22424242422222···22···2

81 irreducible representations

dim1111112222222222222222
type++++++++-+-
imageC1C2C2C3C6C6S3D4D6D8C3×S3D12C3×D4Q32S3×C6D24C3×D8C3×D12Dic24C3×Q32C3×D24C3×Dic24
kernelC3×Dic24C3×C48C3×Dic12Dic24C48Dic12C48C3×C12C24C3×C6C16C12C12C32C8C6C6C4C3C3C2C1
# reps11222411122224244488816

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

350
035
,
310
072
,
01
960
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
 

Export

Subgroup lattice of C3×Dic24 in TeX

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