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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×SD16
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×SD16 — C6×SD16
 Lower central C1 — C2 — C4 — C6×SD16
 Upper central C1 — C2×C6 — C2×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, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×SD16, C2×C24, C3×SD16, C6×D4, C6×Q8, C6×SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C3×D4, C22×C6, C2×SD16, C3×SD16, C6×D4, C6×SD16

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 4 4 1 1 2 2 4 4 1 ··· 1 4 4 4 4 2 2 2 2 2 2 2 2 4 4 4 4 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 SD16 C3×D4 C3×D4 C3×SD16 kernel C6×SD16 C2×C24 C3×SD16 C6×D4 C6×Q8 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C12 C2×C6 C6 C4 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 4 2 2 8

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

 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 6 0 0 67 67
,
 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
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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