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## G = D4×C3×C6order 144 = 24·32

### Direct product of C3×C6 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C3×C6
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32 — D4×C3×C6
 Lower central C1 — C2 — D4×C3×C6
 Upper central C1 — C62 — D4×C3×C6

Generators and relations for D4×C3×C6
G = < a,b,c,d | a3=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 210 in 162 conjugacy classes, 114 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×4], C4 [×2], C22, C22 [×4], C22 [×4], C6 [×12], C6 [×16], C2×C4, D4 [×4], C23 [×2], C32, C12 [×8], C2×C6 [×20], C2×C6 [×16], C2×D4, C3×C6, C3×C6 [×2], C3×C6 [×4], C2×C12 [×4], C3×D4 [×16], C22×C6 [×8], C3×C12 [×2], C62, C62 [×4], C62 [×4], C6×D4 [×4], C6×C12, D4×C32 [×4], C2×C62 [×2], D4×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C22 [×7], C6 [×28], D4 [×2], C23, C32, C2×C6 [×28], C2×D4, C3×C6 [×7], C3×D4 [×8], C22×C6 [×4], C62 [×7], C6×D4 [×4], D4×C32 [×2], C2×C62, D4×C3×C6

Smallest permutation representation of D4×C3×C6
On 72 points
Generators in S72
(1 40 35)(2 41 36)(3 42 31)(4 37 32)(5 38 33)(6 39 34)(7 17 28)(8 18 29)(9 13 30)(10 14 25)(11 15 26)(12 16 27)(19 71 50)(20 72 51)(21 67 52)(22 68 53)(23 69 54)(24 70 49)(43 56 64)(44 57 65)(45 58 66)(46 59 61)(47 60 62)(48 55 63)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 58 10 50)(2 59 11 51)(3 60 12 52)(4 55 7 53)(5 56 8 54)(6 57 9 49)(13 24 39 65)(14 19 40 66)(15 20 41 61)(16 21 42 62)(17 22 37 63)(18 23 38 64)(25 71 35 45)(26 72 36 46)(27 67 31 47)(28 68 32 48)(29 69 33 43)(30 70 34 44)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 50)(8 51)(9 52)(10 53)(11 54)(12 49)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 44)(32 45)(33 46)(34 47)(35 48)(36 43)(37 66)(38 61)(39 62)(40 63)(41 64)(42 65)

G:=sub<Sym(72)| (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,71,50)(20,72,51)(21,67,52)(22,68,53)(23,69,54)(24,70,49)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,58,10,50)(2,59,11,51)(3,60,12,52)(4,55,7,53)(5,56,8,54)(6,57,9,49)(13,24,39,65)(14,19,40,66)(15,20,41,61)(16,21,42,62)(17,22,37,63)(18,23,38,64)(25,71,35,45)(26,72,36,46)(27,67,31,47)(28,68,32,48)(29,69,33,43)(30,70,34,44), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,50)(8,51)(9,52)(10,53)(11,54)(12,49)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65)>;

G:=Group( (1,40,35)(2,41,36)(3,42,31)(4,37,32)(5,38,33)(6,39,34)(7,17,28)(8,18,29)(9,13,30)(10,14,25)(11,15,26)(12,16,27)(19,71,50)(20,72,51)(21,67,52)(22,68,53)(23,69,54)(24,70,49)(43,56,64)(44,57,65)(45,58,66)(46,59,61)(47,60,62)(48,55,63), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,58,10,50)(2,59,11,51)(3,60,12,52)(4,55,7,53)(5,56,8,54)(6,57,9,49)(13,24,39,65)(14,19,40,66)(15,20,41,61)(16,21,42,62)(17,22,37,63)(18,23,38,64)(25,71,35,45)(26,72,36,46)(27,67,31,47)(28,68,32,48)(29,69,33,43)(30,70,34,44), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,50)(8,51)(9,52)(10,53)(11,54)(12,49)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,44)(32,45)(33,46)(34,47)(35,48)(36,43)(37,66)(38,61)(39,62)(40,63)(41,64)(42,65) );

G=PermutationGroup([(1,40,35),(2,41,36),(3,42,31),(4,37,32),(5,38,33),(6,39,34),(7,17,28),(8,18,29),(9,13,30),(10,14,25),(11,15,26),(12,16,27),(19,71,50),(20,72,51),(21,67,52),(22,68,53),(23,69,54),(24,70,49),(43,56,64),(44,57,65),(45,58,66),(46,59,61),(47,60,62),(48,55,63)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,58,10,50),(2,59,11,51),(3,60,12,52),(4,55,7,53),(5,56,8,54),(6,57,9,49),(13,24,39,65),(14,19,40,66),(15,20,41,61),(16,21,42,62),(17,22,37,63),(18,23,38,64),(25,71,35,45),(26,72,36,46),(27,67,31,47),(28,68,32,48),(29,69,33,43),(30,70,34,44)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,50),(8,51),(9,52),(10,53),(11,54),(12,49),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20),(25,68),(26,69),(27,70),(28,71),(29,72),(30,67),(31,44),(32,45),(33,46),(34,47),(35,48),(36,43),(37,66),(38,61),(39,62),(40,63),(41,64),(42,65)])

D4×C3×C6 is a maximal subgroup of
C62.116D4  (C6×D4).S3  C62.38D4  C62.131D4  C62.72D4  C62.254C23  C6213D4  C62.256C23  C6214D4  C62.258C23  C3282+ 1+4

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 4A 4B 6A ··· 6X 6Y ··· 6BD 12A ··· 12P order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 1 ··· 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2

90 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×C3×C6 C6×C12 D4×C32 C2×C62 C6×D4 C2×C12 C3×D4 C22×C6 C3×C6 C6 # reps 1 1 4 2 8 8 32 16 2 16

Matrix representation of D4×C3×C6 in GL3(𝔽13) generated by

 3 0 0 0 9 0 0 0 9
,
 12 0 0 0 4 0 0 0 4
,
 12 0 0 0 12 11 0 1 1
,
 12 0 0 0 12 11 0 0 1
G:=sub<GL(3,GF(13))| [3,0,0,0,9,0,0,0,9],[12,0,0,0,4,0,0,0,4],[12,0,0,0,12,1,0,11,1],[12,0,0,0,12,0,0,11,1] >;

D4×C3×C6 in GAP, Magma, Sage, TeX

D_4\times C_3\times C_6
% in TeX

G:=Group("D4xC3xC6");
// GroupNames label

G:=SmallGroup(144,179);
// by ID

G=gap.SmallGroup(144,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-2,889]);
// Polycyclic

G:=Group<a,b,c,d|a^3=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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