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## G = C6×D8order 96 = 25·3

### Direct product of C6 and D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×D8
 Chief series C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C6×D8
 Lower central C1 — C2 — C4 — C6×D8
 Upper central C1 — C2×C6 — C2×C12 — C6×D8

Generators and relations for C6×D8
G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C6, C6 [×2], C6 [×4], C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C12 [×2], C2×C6, C2×C6 [×8], C2×C8, D8 [×4], C2×D4 [×2], C24 [×2], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×C6 [×2], C2×D8, C2×C24, C3×D8 [×4], C6×D4 [×2], C6×D8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], D8 [×2], C2×D4, C3×D4 [×2], C22×C6, C2×D8, C3×D8 [×2], C6×D4, C6×D8

Smallest permutation representation of C6×D8
On 48 points
Generators in S48
(1 14 39 17 43 30)(2 15 40 18 44 31)(3 16 33 19 45 32)(4 9 34 20 46 25)(5 10 35 21 47 26)(6 11 36 22 48 27)(7 12 37 23 41 28)(8 13 38 24 42 29)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)

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

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

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

C6×D8 is a maximal subgroup of
D81Dic3  D8.Dic3  D8.D6  Dic3⋊D8  C245D4  D8⋊Dic3  (C6×D8).C2  C2411D4  C24.22D4  D12⋊D4  D63D8  Dic6⋊D4  C2412D4  C24.23D4  D813D6

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 D8 C3×D4 C3×D4 C3×D8 kernel C6×D8 C2×C24 C3×D8 C6×D4 C2×D8 C2×C8 D8 C2×D4 C12 C2×C6 C6 C4 C22 C2 # reps 1 1 4 2 2 2 8 4 1 1 4 2 2 8

Matrix representation of C6×D8 in GL3(𝔽73) generated by

 72 0 0 0 64 0 0 0 64
,
 72 0 0 0 41 57 0 32 0
,
 1 0 0 0 0 57 0 41 0
G:=sub<GL(3,GF(73))| [72,0,0,0,64,0,0,0,64],[72,0,0,0,41,32,0,57,0],[1,0,0,0,0,41,0,57,0] >;

C6×D8 in GAP, Magma, Sage, TeX

C_6\times D_8
% in TeX

G:=Group("C6xD8");
// GroupNames label

G:=SmallGroup(96,179);
// by ID

G=gap.SmallGroup(96,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,2164,1090,88]);
// 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^-1>;
// generators/relations

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