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G = C3×SD32order 96 = 25·3

Direct product of C3 and SD32

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

Aliases: C3×SD32, D8.C6, C484C2, C162C6, Q161C6, C6.16D8, C12.37D4, C24.20C22, C8.3(C2×C6), C4.2(C3×D4), C2.4(C3×D8), (C3×Q16)⋊5C2, (C3×D8).2C2, SmallGroup(96,62)

Series: Derived Chief Lower central Upper central

C1C8 — C3×SD32
C1C2C4C8C24C3×Q16 — C3×SD32
C1C2C4C8 — C3×SD32
C1C6C12C24 — C3×SD32

Generators and relations for C3×SD32
 G = < a,b,c | a3=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

8C2
4C22
4C4
8C6
2D4
2Q8
4C12
4C2×C6
2C3×Q8
2C3×D4

Smallest permutation representation of C3×SD32
On 48 points
Generators in S48
(1 18 34)(2 19 35)(3 20 36)(4 21 37)(5 22 38)(6 23 39)(7 24 40)(8 25 41)(9 26 42)(10 27 43)(11 28 44)(12 29 45)(13 30 46)(14 31 47)(15 32 48)(16 17 33)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 27)(19 25)(20 32)(21 23)(22 30)(24 28)(29 31)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)

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

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

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

C3×SD32 is a maximal subgroup of   D48⋊C2  SD32⋊S3  D6.2D8

33 conjugacy classes

class 1 2A2B3A3B4A4B6A6B6C6D8A8B12A12B12C12D16A16B16C16D24A24B24C24D48A···48H
order122334466668812121212161616162424242448···48
size11811281188222288222222222···2

33 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C3C6C6C6D4D8C3×D4SD32C3×D8C3×SD32
kernelC3×SD32C48C3×D8C3×Q16SD32C16D8Q16C12C6C4C3C2C1
# reps11112222122448

Matrix representation of C3×SD32 in GL2(𝔽7) generated by

20
02
,
01
16
,
16
06
G:=sub<GL(2,GF(7))| [2,0,0,2],[0,1,1,6],[1,0,6,6] >;

C3×SD32 in GAP, Magma, Sage, TeX

C_3\times {\rm SD}_{32}
% in TeX

G:=Group("C3xSD32");
// GroupNames label

G:=SmallGroup(96,62);
// by ID

G=gap.SmallGroup(96,62);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,288,169,867,441,165,2164,1090,88]);
// 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^7>;
// generators/relations

Export

Subgroup lattice of C3×SD32 in TeX

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