Copied to
clipboard

G = C3×C48⋊C2order 288 = 25·32

Direct product of C3 and C48⋊C2

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C48⋊C2, C482C6, C484S3, D24.1C6, C6.20D24, C24.80D6, C325SD32, Dic121C6, C12.63D12, C162(C3×S3), (C3×C48)⋊4C2, C6.2(C3×D8), C8.14(S3×C6), C31(C3×SD32), C2.4(C3×D24), C4.2(C3×D12), (C3×C6).19D8, C24.17(C2×C6), (C3×D24).1C2, C12.25(C3×D4), (C3×Dic12)⋊4C2, (C3×C12).127D4, (C3×C24).55C22, SmallGroup(288,234)

Series: Derived Chief Lower central Upper central

C1C24 — C3×C48⋊C2
C1C3C6C12C24C3×C24C3×D24 — C3×C48⋊C2
C3C6C12C24 — C3×C48⋊C2
C1C6C12C24C48

Generators and relations for C3×C48⋊C2
 G = < a,b,c | a3=b48=c2=1, ab=ba, ac=ca, cbc=b23 >

Subgroups: 214 in 57 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24 [×2], C24, Dic6, D12, C3×D4, C3×Q8, SD32, C3×Dic3, C3×C12, S3×C6, C48 [×2], C48, D24, Dic12, C3×D8, C3×Q16, C3×C24, C3×Dic6, C3×D12, C48⋊C2, C3×SD32, C3×C48, C3×D24, C3×Dic12, C3×C48⋊C2
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, D12, C3×D4, SD32, S3×C6, D24, C3×D8, C3×D12, C48⋊C2, C3×SD32, C3×D24, C3×C48⋊C2

Smallest permutation representation of C3×C48⋊C2
On 96 points
Generators in S96
(1 17 33)(2 18 34)(3 19 35)(4 20 36)(5 21 37)(6 22 38)(7 23 39)(8 24 40)(9 25 41)(10 26 42)(11 27 43)(12 28 44)(13 29 45)(14 30 46)(15 31 47)(16 32 48)(49 81 65)(50 82 66)(51 83 67)(52 84 68)(53 85 69)(54 86 70)(55 87 71)(56 88 72)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(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 64)(2 87)(3 62)(4 85)(5 60)(6 83)(7 58)(8 81)(9 56)(10 79)(11 54)(12 77)(13 52)(14 75)(15 50)(16 73)(17 96)(18 71)(19 94)(20 69)(21 92)(22 67)(23 90)(24 65)(25 88)(26 63)(27 86)(28 61)(29 84)(30 59)(31 82)(32 57)(33 80)(34 55)(35 78)(36 53)(37 76)(38 51)(39 74)(40 49)(41 72)(42 95)(43 70)(44 93)(45 68)(46 91)(47 66)(48 89)

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

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

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

81 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A···12H12I12J16A16B16C16D24A···24P48A···48AF
order122333334466666668812···1212121616161624···2448···48
size112411222224112222424222···2242422222···22···2

81 irreducible representations

dim111111112222222222222222
type++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D8C3×S3D12C3×D4SD32S3×C6D24C3×D8C3×D12C48⋊C2C3×SD32C3×D24C3×C48⋊C2
kernelC3×C48⋊C2C3×C48C3×D24C3×Dic12C48⋊C2C48D24Dic12C48C3×C12C24C3×C6C16C12C12C32C8C6C6C4C3C3C2C1
# reps1111222211122224244488816

Matrix representation of C3×C48⋊C2 in GL2(𝔽97) generated by

610
061
,
950
2149
,
1434
7783
G:=sub<GL(2,GF(97))| [61,0,0,61],[95,21,0,49],[14,77,34,83] >;

C3×C48⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{48}\rtimes C_2
% in TeX

G:=Group("C3xC48:C2");
// GroupNames label

G:=SmallGroup(288,234);
// by ID

G=gap.SmallGroup(288,234);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,260,2355,80,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^23>;
// generators/relations

׿
×
𝔽