Copied to
clipboard

## 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

 Derived series C1 — C8 — C3×D16
 Chief series C1 — C2 — C4 — C8 — C24 — C3×D8 — C3×D16
 Lower central C1 — C2 — C4 — C8 — C3×D16
 Upper central C1 — C6 — C12 — C24 — C3×D16

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

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 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 16A 16B 16C 16D 24A 24B 24C 24D 48A ··· 48H order 1 2 2 2 3 3 4 6 6 6 6 6 6 8 8 12 12 16 16 16 16 24 24 24 24 48 ··· 48 size 1 1 8 8 1 1 2 1 1 8 8 8 8 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 D16 C3×D8 C3×D16 kernel C3×D16 C48 C3×D8 D16 C16 D8 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 8

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

 25 0 0 25
,
 14 25 26 0
,
 0 25 5 0
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

׿
×
𝔽