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## G = C6×C4○D4order 96 = 25·3

### Direct product of C6 and C4○D4

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

Aliases: C6×C4○D4, C6.18C24, C12.55C23, C4(C6×D4), C4(C6×Q8), C12(C2×D4), C12(C6×D4), D4(C2×C12), C12(C2×Q8), C12(C6×Q8), Q8(C2×C12), (C2×D4)⋊7C6, D43(C2×C6), (C2×Q8)⋊8C6, Q84(C2×C6), C12(C4○D4), (C6×D4)⋊16C2, (C22×C4)⋊8C6, (C6×Q8)⋊13C2, C4.8(C22×C6), C2.3(C23×C6), (C2×C6).6C23, (C22×C12)⋊13C2, (C2×C12)⋊16C22, (C3×D4)⋊12C22, C23.14(C2×C6), (C3×Q8)⋊11C22, C22.1(C22×C6), (C22×C6).30C22, C4(C3×C4○D4), (C2×C4)(C3×D4), (C2×C4)(C3×Q8), (C2×C4)(C6×Q8), (C2×C4)⋊5(C2×C6), C12(C3×C4○D4), (C2×C12)(C3×D4), (C2×C12)(C3×Q8), (C2×Q8)(C2×C12), (C2×C12)(C6×Q8), SmallGroup(96,223)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C4○D4
G = < a,b,c,d | a6=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C6×C4○D4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, C3×C4○D4, C23×C6, C6×C4○D4

Smallest permutation representation of C6×C4○D4
On 48 points
Generators in S48
(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 29 18 19)(2 30 13 20)(3 25 14 21)(4 26 15 22)(5 27 16 23)(6 28 17 24)(7 33 48 37)(8 34 43 38)(9 35 44 39)(10 36 45 40)(11 31 46 41)(12 32 47 42)
(1 34 18 38)(2 35 13 39)(3 36 14 40)(4 31 15 41)(5 32 16 42)(6 33 17 37)(7 28 48 24)(8 29 43 19)(9 30 44 20)(10 25 45 21)(11 26 46 22)(12 27 47 23)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(13 42)(14 37)(15 38)(16 39)(17 40)(18 41)(25 48)(26 43)(27 44)(28 45)(29 46)(30 47)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C4○D4 C3×C4○D4 kernel C6×C4○D4 C22×C12 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C6 C2 # reps 1 3 3 1 8 2 6 6 2 16 4 8

Matrix representation of C6×C4○D4 in GL4(𝔽13) generated by

 12 0 0 0 0 3 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 0 5
,
 12 0 0 0 0 1 0 0 0 0 1 11 0 0 1 12
,
 12 0 0 0 0 12 0 0 0 0 1 11 0 0 0 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,3,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,1,0,0,0,0,1,1,0,0,11,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,11,12] >;

C6×C4○D4 in GAP, Magma, Sage, TeX

C_6\times C_4\circ D_4
% in TeX

G:=Group("C6xC4oD4");
// GroupNames label

G:=SmallGroup(96,223);
// by ID

G=gap.SmallGroup(96,223);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,230]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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