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## G = C3×C48⋊C2order 288 = 25·32

### Direct product of C3 and C48⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C48⋊C2
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×D24 — C3×C48⋊C2
 Lower central C3 — C6 — C12 — C24 — C3×C48⋊C2
 Upper central C1 — C6 — C12 — C24 — C48

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

81 irreducible representations

 dim 1 1 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 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 D12 C3×D4 SD32 S3×C6 D24 C3×D8 C3×D12 C48⋊C2 C3×SD32 C3×D24 C3×C48⋊C2 kernel C3×C48⋊C2 C3×C48 C3×D24 C3×Dic12 C48⋊C2 C48 D24 Dic12 C48 C3×C12 C24 C3×C6 C16 C12 C12 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 4 8 8 8 16

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

 61 0 0 61
,
 95 0 21 49
,
 14 34 77 83
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

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