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G = C2×C48order 96 = 25·3

Abelian group of type [2,48]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C48, SmallGroup(96,59)

Series: Derived Chief Lower central Upper central

C1 — C2×C48
C1C2C4C8C24C48 — C2×C48
C1 — C2×C48
C1 — C2×C48

Generators and relations for C2×C48
 G = < a,b | a2=b48=1, ab=ba >


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

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

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

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

C2×C48 is a maximal subgroup of
C3⋊M6(2)  Dic3⋊C16  C4810C4  C2.Dic24  C485C4  C486C4  C48.C4  D6⋊C16  D12.C8  C2.D48  D24.1C4  D12.4C8  D487C2

96 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F8A···8H12A···12H16A···16P24A···24P48A···48AF
order12223344446···68···812···1216···1624···2448···48
size11111111111···11···11···11···11···11···1

96 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C24C24C48
kernelC2×C48C48C2×C24C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps121222424444168832

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

960
01
,
180
03
G:=sub<GL(2,GF(97))| [96,0,0,1],[18,0,0,3] >;

C2×C48 in GAP, Magma, Sage, TeX

C_2\times C_{48}
% in TeX

G:=Group("C2xC48");
// GroupNames label

G:=SmallGroup(96,59);
// by ID

G=gap.SmallGroup(96,59);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,69,88]);
// Polycyclic

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

Export

Subgroup lattice of C2×C48 in TeX

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