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G = C2×D48  order 192 = 26·3

Direct product of C2 and D48

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D48, C16⋊7D6, C6⋊1D16, C4.6D24, C48⋊8C22, C24.60D4, C8.10D12, C12.31D8, D24⋊7C22, C24.55C23, C22.12D24, C3⋊1(C2×D16), (C2×C16)⋊5S3, (C2×C48)⋊9C2, C6.9(C2×D8), (C2×D24)⋊8C2, (C2×C6).18D8, (C2×C8).303D6, C4.36(C2×D12), C2.11(C2×D24), (C2×C4).83D12, (C2×C12).380D4, C12.279(C2×D4), C8.45(C22×S3), (C2×C24).376C22, SmallGroup(192,461)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C2×D48
Chief series
C1 — C3 — C6 — C12 — C24 — D24 — C2×D24 — C2×D48
Lower central
C3 — C6 — C12 — C24 — C2×D48
Upper central
C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×D48
 G = < a,b,c | a2=b48=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 520 in 98 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C12, D6, C2×C6, C16, C2×C8, D8, C2×D4, C24, D12, C2×C12, C22×S3, C2×C16, D16, C2×D8, C48, D24, D24, C2×C24, C2×D12, C2×D16, D48, C2×C48, C2×D24, C2×D48
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, D16, C2×D8, D24, C2×D12, C2×D16, D48, C2×D24, C2×D48

Smallest permutation representation of C2×D48
►On 96 points
Generators in S96
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B6A6B6C8A8B8C8D12A12B12C12D16A···16H24A···24H48A···48P
order1222222234466688881212121216···1624···2448···48
size111124242424222222222222222···22···22···2

54 irreducible representations

dim11112222222222222
type+++++++++++++++++
imageC1C2C2C2S3D4D4D6D6D8D8D12D12D16D24D24D48
kernelC2×D48D48C2×C48C2×D24C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps141211121222284416

Matrix representation of C2×D48 ►in GL3(𝔽97) generated by

9600
010
001
,
9600
01965
03284
,
9600
03284
01965
G:=sub<GL(3,GF(97))| [96,0,0,0,1,0,0,0,1],[96,0,0,0,19,32,0,65,84],[96,0,0,0,32,19,0,84,65] >;
 

C2×D48 in GAP, Magma, Sage, TeX 

C_2\times D_{48}
 
% in TeX
 
G:=Group("C2xD48");
 
// GroupNames label
 
G:=SmallGroup(192,461);
 
// by ID
 
G=gap.SmallGroup(192,461);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,142,675,192,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

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