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## G = Dic3×C2×C6order 144 = 24·32

### Direct product of C2×C6 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C6
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C6×Dic3 — Dic3×C2×C6
 Lower central C3 — Dic3×C2×C6
 Upper central C1 — C22×C6

Generators and relations for Dic3×C2×C6
G = < a,b,c,d | a2=b6=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 184 in 124 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×4], C22 [×7], C6 [×2], C6 [×12], C6 [×7], C2×C4 [×6], C23, C32, Dic3 [×4], C12 [×4], C2×C6 [×14], C2×C6 [×7], C22×C4, C3×C6, C3×C6 [×6], C2×Dic3 [×6], C2×C12 [×6], C22×C6 [×2], C22×C6, C3×Dic3 [×4], C62 [×7], C22×Dic3, C22×C12, C6×Dic3 [×6], C2×C62, Dic3×C2×C6
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C3×Dic3 [×4], S3×C6 [×3], C22×Dic3, C22×C12, C6×Dic3 [×6], S3×C2×C6, Dic3×C2×C6

Smallest permutation representation of Dic3×C2×C6
On 48 points
Generators in S48
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 29)(14 30)(15 25)(16 26)(17 27)(18 28)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)
(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 13 3 15 5 17)(2 14 4 16 6 18)(7 48 11 46 9 44)(8 43 12 47 10 45)(19 28 21 30 23 26)(20 29 22 25 24 27)(31 38 35 42 33 40)(32 39 36 37 34 41)
(1 44 15 11)(2 45 16 12)(3 46 17 7)(4 47 18 8)(5 48 13 9)(6 43 14 10)(19 42 30 31)(20 37 25 32)(21 38 26 33)(22 39 27 34)(23 40 28 35)(24 41 29 36)

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

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

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

Dic3×C2×C6 is a maximal subgroup of
C62.6Q8  C62.94C23  C62.97C23  C62.56D4  C623Q8  C62.60D4  C626D4  S3×C22×C12

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 kernel Dic3×C2×C6 C6×Dic3 C2×C62 C22×Dic3 C62 C2×Dic3 C22×C6 C2×C6 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 8 12 2 16 1 4 3 2 8 6

Matrix representation of Dic3×C2×C6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 9 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 12 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 5 0 0 0 0 12 0 0 0 0 0 12 0 0 12 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[12,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[5,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0] >;

Dic3×C2×C6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_6
% in TeX

G:=Group("Dic3xC2xC6");
// GroupNames label

G:=SmallGroup(144,166);
// by ID

G=gap.SmallGroup(144,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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