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G = C4×C24order 96 = 25·3

Abelian group of type [4,24]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C24, SmallGroup(96,46)

Series: Derived Chief Lower central Upper central

C1 — C4×C24
C1C2C22C2×C4C2×C12C2×C24 — C4×C24
C1 — C4×C24
C1 — C4×C24

Generators and relations for C4×C24
 G = < a,b | a4=b24=1, ab=ba >


Smallest permutation representation of C4×C24
Regular action on 96 points
Generators in S96
(1 65 28 78)(2 66 29 79)(3 67 30 80)(4 68 31 81)(5 69 32 82)(6 70 33 83)(7 71 34 84)(8 72 35 85)(9 49 36 86)(10 50 37 87)(11 51 38 88)(12 52 39 89)(13 53 40 90)(14 54 41 91)(15 55 42 92)(16 56 43 93)(17 57 44 94)(18 58 45 95)(19 59 46 96)(20 60 47 73)(21 61 48 74)(22 62 25 75)(23 63 26 76)(24 64 27 77)
(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)| (1,65,28,78)(2,66,29,79)(3,67,30,80)(4,68,31,81)(5,69,32,82)(6,70,33,83)(7,71,34,84)(8,72,35,85)(9,49,36,86)(10,50,37,87)(11,51,38,88)(12,52,39,89)(13,53,40,90)(14,54,41,91)(15,55,42,92)(16,56,43,93)(17,57,44,94)(18,58,45,95)(19,59,46,96)(20,60,47,73)(21,61,48,74)(22,62,25,75)(23,63,26,76)(24,64,27,77), (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( (1,65,28,78)(2,66,29,79)(3,67,30,80)(4,68,31,81)(5,69,32,82)(6,70,33,83)(7,71,34,84)(8,72,35,85)(9,49,36,86)(10,50,37,87)(11,51,38,88)(12,52,39,89)(13,53,40,90)(14,54,41,91)(15,55,42,92)(16,56,43,93)(17,57,44,94)(18,58,45,95)(19,59,46,96)(20,60,47,73)(21,61,48,74)(22,62,25,75)(23,63,26,76)(24,64,27,77), (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([(1,65,28,78),(2,66,29,79),(3,67,30,80),(4,68,31,81),(5,69,32,82),(6,70,33,83),(7,71,34,84),(8,72,35,85),(9,49,36,86),(10,50,37,87),(11,51,38,88),(12,52,39,89),(13,53,40,90),(14,54,41,91),(15,55,42,92),(16,56,43,93),(17,57,44,94),(18,58,45,95),(19,59,46,96),(20,60,47,73),(21,61,48,74),(22,62,25,75),(23,63,26,76),(24,64,27,77)], [(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)])

C4×C24 is a maximal subgroup of
C42.279D6  C24⋊C8  C4.8Dic12  C242C8  C241C8  C4.17D24  C24.C8  C12⋊C16  C24.1C8  C2412Q8  C249Q8  C12.14Q16  C248Q8  C24.13Q8  C42.282D6  C86D12  D6.C42  C42.243D6  C85D12  C4.5D24  C124D8  C8.8D12  C42.264D6  C124Q16  D2411C4

96 conjugacy classes

class 1 2A2B2C3A3B4A···4L6A···6F8A···8P12A···12X24A···24AF
order1222334···46···68···812···1224···24
size1111111···11···11···11···11···1

96 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C12C12C24
kernelC4×C24C4×C12C2×C24C4×C8C24C2×C12C42C2×C8C12C8C2×C4C4
# reps112284241616832

Matrix representation of C4×C24 in GL2(𝔽73) generated by

460
072
,
510
021
G:=sub<GL(2,GF(73))| [46,0,0,72],[51,0,0,21] >;

C4×C24 in GAP, Magma, Sage, TeX

C_4\times C_{24}
% in TeX

G:=Group("C4xC24");
// GroupNames label

G:=SmallGroup(96,46);
// by ID

G=gap.SmallGroup(96,46);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,72,151,117]);
// Polycyclic

G:=Group<a,b|a^4=b^24=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C24 in TeX

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