Copied to
clipboard

G = Dic24order 96 = 25·3

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic24, C16.S3, C31Q32, C6.3D8, C48.1C2, C4.3D12, C2.5D24, C8.15D6, C12.26D4, C24.16C22, Dic12.1C2, SmallGroup(96,8)

Series: Derived Chief Lower central Upper central

C1C24 — Dic24
C1C3C6C12C24Dic12 — Dic24
C3C6C12C24 — Dic24
C1C2C4C8C16

Generators and relations for Dic24
 G = < a,b | a48=1, b2=a24, bab-1=a-1 >

12C4
12C4
6Q8
6Q8
4Dic3
4Dic3
3Q16
3Q16
2Dic6
2Dic6
3Q32

Character table of Dic24

 class 1234A4B4C68A8B12A12B16A16B16C16D24A24B24C24D48A48B48C48D48E48F48G48H
 size 11222424222222222222222222222
ρ1111111111111111111111111111    trivial
ρ21111-1111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111111111111111111111    linear of order 2
ρ522-1200-122-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62222002-2-2220000-2-2-2-200000000    orthogonal lifted from D4
ρ722-1200-122-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ8222-200200-2-22-2-220000-222-222-2-2    orthogonal lifted from D8
ρ9222-200200-2-2-222-200002-2-22-2-222    orthogonal lifted from D8
ρ1022-1200-1-2-2-1-100001111-33-333-33-3    orthogonal lifted from D12
ρ1122-1200-1-2-2-1-1000011113-33-3-33-33    orthogonal lifted from D12
ρ1222-1-200-10011-222-2-333-3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3    orthogonal lifted from D24
ρ1322-1-200-10011-222-23-3-33ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385    orthogonal lifted from D24
ρ1422-1-200-100112-2-22-333-3ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328    orthogonal lifted from D24
ρ1522-1-200-100112-2-223-3-33ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32    orthogonal lifted from D24
ρ162-22000-22-200165163ζ16151691615169ζ165163-2-222ζ1615169ζ165163ζ165163ζ161516916516316516316151691615169    symplectic lifted from Q32, Schur index 2
ρ172-22000-2-2200ζ1615169ζ165163165163161516922-2-2ζ16516316151691615169ζ165163ζ1615169ζ1615169165163165163    symplectic lifted from Q32, Schur index 2
ρ182-22000-22-200ζ1651631615169ζ1615169165163-2-22216151691651631651631615169ζ165163ζ165163ζ1615169ζ1615169    symplectic lifted from Q32, Schur index 2
ρ192-22000-2-22001615169165163ζ165163ζ161516922-2-2165163ζ1615169ζ161516916516316151691615169ζ165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ202-2-100012-2-33ζ165163161516916716165163ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ167ζ3216ζ3216ζ1613ζ321611ζ321611ζ165ζ32165163ζ32ζ167ζ316ζ316ζ1613ζ316131611ζ3ζ165ζ3163ζ3163ζ167ζ3216716ζ32ζ167ζ316716ζ3    symplectic faithful, Schur index 2
ρ212-2-100012-23-3ζ165163161516916716165163ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ167ζ316ζ316ζ165ζ32165163ζ32ζ1613ζ321611ζ321611ζ167ζ3216ζ3216ζ165ζ3163ζ3163ζ1613ζ316131611ζ3ζ167ζ316716ζ3ζ167ζ3216716ζ32    symplectic faithful, Schur index 2
ρ222-2-100012-23-3165163167161615169ζ165163ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ167ζ3216716ζ32ζ165ζ3163ζ3163ζ1613ζ316131611ζ3ζ167ζ316716ζ3ζ165ζ32165163ζ32ζ1613ζ321611ζ321611ζ167ζ3216ζ3216ζ167ζ316ζ316    symplectic faithful, Schur index 2
ρ232-2-10001-22-331615169165163ζ16516316716ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ165ζ32165163ζ32ζ167ζ3216716ζ32ζ167ζ316716ζ3ζ1613ζ321611ζ321611ζ167ζ316ζ316ζ167ζ3216ζ3216ζ1613ζ316131611ζ3ζ165ζ3163ζ3163    symplectic faithful, Schur index 2
ρ242-2-10001-223-31615169165163ζ16516316716ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ1613ζ321611ζ321611ζ167ζ316716ζ3ζ167ζ3216716ζ32ζ165ζ32165163ζ32ζ167ζ3216ζ3216ζ167ζ316ζ316ζ165ζ3163ζ3163ζ1613ζ316131611ζ3    symplectic faithful, Schur index 2
ρ252-2-10001-223-316716ζ1651631651631615169ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ1613ζ316131611ζ3ζ167ζ3216ζ3216ζ167ζ316ζ316ζ165ζ3163ζ3163ζ167ζ316716ζ3ζ167ζ3216716ζ32ζ165ζ32165163ζ32ζ1613ζ321611ζ321611    symplectic faithful, Schur index 2
ρ262-2-10001-22-3316716ζ1651631651631615169ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ165ζ3163ζ3163ζ167ζ316ζ316ζ167ζ3216ζ3216ζ1613ζ316131611ζ3ζ167ζ3216716ζ32ζ167ζ316716ζ3ζ1613ζ321611ζ321611ζ165ζ32165163ζ32    symplectic faithful, Schur index 2
ρ272-2-100012-2-33165163167161615169ζ165163ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ167ζ316716ζ3ζ1613ζ316131611ζ3ζ165ζ3163ζ3163ζ167ζ3216716ζ32ζ1613ζ321611ζ321611ζ165ζ32165163ζ32ζ167ζ316ζ316ζ167ζ3216ζ3216    symplectic faithful, Schur index 2

Smallest permutation representation of Dic24
Regular action on 96 points
Generators in S96
(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 77 25 53)(2 76 26 52)(3 75 27 51)(4 74 28 50)(5 73 29 49)(6 72 30 96)(7 71 31 95)(8 70 32 94)(9 69 33 93)(10 68 34 92)(11 67 35 91)(12 66 36 90)(13 65 37 89)(14 64 38 88)(15 63 39 87)(16 62 40 86)(17 61 41 85)(18 60 42 84)(19 59 43 83)(20 58 44 82)(21 57 45 81)(22 56 46 80)(23 55 47 79)(24 54 48 78)

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

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

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

Dic24 is a maximal subgroup of
C32⋊S3  Dic48  D16.S3  C3⋊Q64  D487C2  C16.D6  D163S3  SD32⋊S3  S3×Q32  Dic72  C323Q32  C325Q32  C5⋊Dic24  Dic120
Dic24 is a maximal quotient of
C2.Dic24  C485C4  Dic72  C323Q32  C325Q32  C5⋊Dic24  Dic120

Matrix representation of Dic24 in GL2(𝔽47) generated by

3211
1142
,
046
10
G:=sub<GL(2,GF(47))| [32,11,11,42],[0,1,46,0] >;

Dic24 in GAP, Magma, Sage, TeX

{\rm Dic}_{24}
% in TeX

G:=Group("Dic24");
// GroupNames label

G:=SmallGroup(96,8);
// by ID

G=gap.SmallGroup(96,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,79,218,122,579,69,2309]);
// Polycyclic

G:=Group<a,b|a^48=1,b^2=a^24,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic24 in TeX
Character table of Dic24 in TeX

׿
×
𝔽