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## G = D4×C2×C6order 96 = 25·3

### Direct product of C2×C6 and D4

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

Aliases: D4×C2×C6, C246C6, C124C23, C6.16C24, C4⋊(C22×C6), (C23×C6)⋊2C2, C234(C2×C6), (C22×C4)⋊7C6, (C2×C6)⋊2C23, C2.1(C23×C6), (C22×C12)⋊12C2, (C2×C12)⋊15C22, (C22×C6)⋊6C22, C222(C22×C6), (C2×C4)⋊4(C2×C6), SmallGroup(96,221)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C6
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4 — D4×C2×C6
 Lower central C1 — C2 — D4×C2×C6
 Upper central C1 — C22×C6 — D4×C2×C6

Generators and relations for D4×C2×C6
G = < a,b,c,d | a2=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22×D4, C22×C12, C6×D4, C23×C6, D4×C2×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4×C2×C6

Smallest permutation representation of D4×C2×C6
On 48 points
Generators in S48
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(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 10 15 44)(2 11 16 45)(3 12 17 46)(4 7 18 47)(5 8 13 48)(6 9 14 43)(19 38 26 31)(20 39 27 32)(21 40 28 33)(22 41 29 34)(23 42 30 35)(24 37 25 36)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 18)(8 13)(9 14)(10 15)(11 16)(12 17)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)

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

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

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

D4×C2×C6 is a maximal subgroup of
(C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  C24.29D6  C24.30D6  C24.31D6  C24.32D6  C24.49D6  C2412D6  C24.52D6  C24.53D6

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A 4B 4C 4D 6A ··· 6N 6O ··· 6AD 12A ··· 12H order 1 2 ··· 2 2 ··· 2 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 ··· 2 1 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel D4×C2×C6 C22×C12 C6×D4 C23×C6 C22×D4 C22×C4 C2×D4 C24 C2×C6 C22 # reps 1 1 12 2 2 2 24 4 4 8

Matrix representation of D4×C2×C6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 8 2 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 8 2 0 0 1 5
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,2,5],[1,0,0,0,0,1,0,0,0,0,8,1,0,0,2,5] >;

D4×C2×C6 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_6
% in TeX

G:=Group("D4xC2xC6");
// GroupNames label

G:=SmallGroup(96,221);
// by ID

G=gap.SmallGroup(96,221);
# by ID

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

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

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