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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

C1C2 — D4×C2×C6
C1C2C6C2×C6C3×D4C6×D4 — D4×C2×C6
C1C2 — D4×C2×C6
C1C22×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 [×6], C2 [×8], C3, C4 [×4], C22 [×15], C22 [×24], C6, C6 [×6], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C12 [×4], C2×C6 [×15], C2×C6 [×24], C22×C4, C2×D4 [×12], C24 [×2], C2×C12 [×6], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×8], C22×D4, C22×C12, C6×D4 [×12], C23×C6 [×2], D4×C2×C6
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C22×D4, C6×D4 [×6], 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 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 10 15 48)(2 11 16 43)(3 12 17 44)(4 7 18 45)(5 8 13 46)(6 9 14 47)(19 40 28 31)(20 41 29 32)(21 42 30 33)(22 37 25 34)(23 38 26 35)(24 39 27 36)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(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)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,10,15,48)(2,11,16,43)(3,12,17,44)(4,7,18,45)(5,8,13,46)(6,9,14,47)(19,40,28,31)(20,41,29,32)(21,42,30,33)(22,37,25,34)(23,38,26,35)(24,39,27,36), (1,48)(2,43)(3,44)(4,45)(5,46)(6,47)(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)(7,36)(8,31)(9,32)(10,33)(11,34)(12,35)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,10,15,48)(2,11,16,43)(3,12,17,44)(4,7,18,45)(5,8,13,46)(6,9,14,47)(19,40,28,31)(20,41,29,32)(21,42,30,33)(22,37,25,34)(23,38,26,35)(24,39,27,36), (1,48)(2,43)(3,44)(4,45)(5,46)(6,47)(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),(7,36),(8,31),(9,32),(10,33),(11,34),(12,35),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,10,15,48),(2,11,16,43),(3,12,17,44),(4,7,18,45),(5,8,13,46),(6,9,14,47),(19,40,28,31),(20,41,29,32),(21,42,30,33),(22,37,25,34),(23,38,26,35),(24,39,27,36)], [(1,48),(2,43),(3,44),(4,45),(5,46),(6,47),(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···2G2H···2O3A3B4A4B4C4D6A···6N6O···6AD12A···12H
order12···22···23344446···66···612···12
size11···12···21122221···12···22···2

60 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C3C6C6C6D4C3×D4
kernelD4×C2×C6C22×C12C6×D4C23×C6C22×D4C22×C4C2×D4C24C2×C6C22
# reps111222224448

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

12000
01200
0010
0001
,
3000
01200
0010
0001
,
12000
01200
0082
0005
,
1000
0100
0082
0015
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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