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G = C3×D16order 96 = 25·3

Direct product of C3 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C3×D16, C483C2, C161C6, D81C6, C6.15D8, C12.36D4, C24.19C22, (C3×D8)⋊5C2, C8.2(C2×C6), C4.1(C3×D4), C2.3(C3×D8), SmallGroup(96,61)

Series: Derived Chief Lower central Upper central

C1C8 — C3×D16
C1C2C4C8C24C3×D8 — C3×D16
C1C2C4C8 — C3×D16
C1C6C12C24 — C3×D16

Generators and relations for C3×D16
 G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

8C2
8C2
4C22
4C22
8C6
8C6
2D4
2D4
4C2×C6
4C2×C6
2C3×D4
2C3×D4

Smallest permutation representation of C3×D16
On 48 points
Generators in S48
(1 32 38)(2 17 39)(3 18 40)(4 19 41)(5 20 42)(6 21 43)(7 22 44)(8 23 45)(9 24 46)(10 25 47)(11 26 48)(12 27 33)(13 28 34)(14 29 35)(15 30 36)(16 31 37)
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 32)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)

G:=sub<Sym(48)| (1,32,38)(2,17,39)(3,18,40)(4,19,41)(5,20,42)(6,21,43)(7,22,44)(8,23,45)(9,24,46)(10,25,47)(11,26,48)(12,27,33)(13,28,34)(14,29,35)(15,30,36)(16,31,37), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,32)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)>;

G:=Group( (1,32,38)(2,17,39)(3,18,40)(4,19,41)(5,20,42)(6,21,43)(7,22,44)(8,23,45)(9,24,46)(10,25,47)(11,26,48)(12,27,33)(13,28,34)(14,29,35)(15,30,36)(16,31,37), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,32)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46) );

G=PermutationGroup([(1,32,38),(2,17,39),(3,18,40),(4,19,41),(5,20,42),(6,21,43),(7,22,44),(8,23,45),(9,24,46),(10,25,47),(11,26,48),(12,27,33),(13,28,34),(14,29,35),(15,30,36),(16,31,37)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(31,32),(33,42),(34,41),(35,40),(36,39),(37,38),(43,48),(44,47),(45,46)])

C3×D16 is a maximal subgroup of   C3⋊D32  D16.S3  D8⋊D6  D163S3

33 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F8A8B12A12B16A16B16C16D24A24B24C24D48A···48H
order1222334666666881212161616162424242448···48
size11881121188882222222222222···2

33 irreducible representations

dim111111222222
type++++++
imageC1C2C2C3C6C6D4D8C3×D4D16C3×D8C3×D16
kernelC3×D16C48C3×D8D16C16D8C12C6C4C3C2C1
# reps112224122448

Matrix representation of C3×D16 in GL2(𝔽31) generated by

250
025
,
1425
260
,
025
50
G:=sub<GL(2,GF(31))| [25,0,0,25],[14,26,25,0],[0,5,25,0] >;

C3×D16 in GAP, Magma, Sage, TeX

C_3\times D_{16}
% in TeX

G:=Group("C3xD16");
// GroupNames label

G:=SmallGroup(96,61);
// by ID

G=gap.SmallGroup(96,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,169,867,441,165,2164,1090,88]);
// Polycyclic

G:=Group<a,b,c|a^3=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D16 in TeX

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