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G = C6×SD16order 96 = 25·3

Direct product of C6 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C6×SD16, C12.42D4, C2413C22, C12.45C23, C83(C2×C6), (C2×C8)⋊5C6, Q82(C2×C6), (C2×Q8)⋊5C6, C4.7(C3×D4), (C2×C24)⋊13C2, (C6×Q8)⋊10C2, (C2×D4).6C6, D4.1(C2×C6), C2.12(C6×D4), (C2×C6).53D4, C6.75(C2×D4), (C6×D4).13C2, C4.2(C22×C6), (C3×Q8)⋊9C22, C22.15(C3×D4), (C3×D4).11C22, (C2×C12).130C22, (C2×C4).26(C2×C6), SmallGroup(96,180)

Series: Derived Chief Lower central Upper central

C1C4 — C6×SD16
C1C2C4C12C3×Q8C3×SD16 — C6×SD16
C1C2C4 — C6×SD16
C1C2×C6C2×C12 — C6×SD16

Generators and relations for C6×SD16
 G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×2], Q8, C23, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C24 [×2], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×C6, C2×SD16, C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C6×SD16
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], SD16 [×2], C2×D4, C3×D4 [×2], C22×C6, C2×SD16, C3×SD16 [×2], C6×D4, C6×SD16

Smallest permutation representation of C6×SD16
On 48 points
Generators in S48
(1 10 34 21 45 26)(2 11 35 22 46 27)(3 12 36 23 47 28)(4 13 37 24 48 29)(5 14 38 17 41 30)(6 15 39 18 42 31)(7 16 40 19 43 32)(8 9 33 20 44 25)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)

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

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

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

C6×SD16 is a maximal subgroup of
Dic33SD16  Dic35SD16  SD16⋊Dic3  (C3×D4).D4  (C3×Q8).D4  C24.31D4  C24.43D4  D66SD16  D68SD16  C2414D4  D127D4  Dic6.16D4  C248D4  C2415D4  C249D4  C24.44D4  SD1613D6

42 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order1222223344446···666668888121212121212121224···24
size1111441122441···144442222222244442···2

42 irreducible representations

dim1111111111222222
type+++++++
imageC1C2C2C2C2C3C6C6C6C6D4D4SD16C3×D4C3×D4C3×SD16
kernelC6×SD16C2×C24C3×SD16C6×D4C6×Q8C2×SD16C2×C8SD16C2×D4C2×Q8C12C2×C6C6C4C22C2
# reps1141122822114228

Matrix representation of C6×SD16 in GL4(𝔽73) generated by

64000
07200
00720
00072
,
1000
0100
00676
006767
,
72000
07200
0010
00072
G:=sub<GL(4,GF(73))| [64,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,67,67,0,0,6,67],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;

C6×SD16 in GAP, Magma, Sage, TeX

C_6\times {\rm SD}_{16}
% in TeX

G:=Group("C6xSD16");
// GroupNames label

G:=SmallGroup(96,180);
// by ID

G=gap.SmallGroup(96,180);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,2164,1090,88]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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