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G = C3×Dic24order 288 = 25·32

Direct product of C3 and Dic24

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic24, C48.1C6, C48.5S3, C324Q32, C6.21D24, C24.81D6, C12.64D12, Dic12.1C6, C16.(C3×S3), C31(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

C1C24 — C3×Dic24
C1C3C6C12C24C3×C24C3×Dic12 — C3×Dic24
C3C6C12C24 — C3×Dic24
C1C6C12C24C48

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

2C3
12C4
12C4
2C6
6Q8
6Q8
2C12
4Dic3
4Dic3
12C12
12C12
3Q16
3Q16
2Dic6
2C24
2Dic6
6C3×Q8
6C3×Q8
4C3×Dic3
4C3×Dic3
3Q32
2C48
3C3×Q16
3C3×Q16
2C3×Dic6
2C3×Dic6
3C3×Q32

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 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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