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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

Aliases: D4×C3×C6, C4⋊C62, C222C62, C626C22, C124(C2×C6), (C2×C12)⋊6C6, (C6×C12)⋊10C2, (C2×C62)⋊1C2, C232(C3×C6), (C22×C6)⋊3C6, C2.1(C2×C62), (C3×C12)⋊10C22, C6.14(C22×C6), (C3×C6).39C23, (C2×C4)⋊2(C3×C6), (C2×C6)⋊4(C2×C6), SmallGroup(144,179)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C3×C6
C1C2C6C3×C6C62D4×C32 — D4×C3×C6
C1C2 — D4×C3×C6
C1C62 — 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 2A2B2C2D2E2F2G3A···3H4A4B6A···6X6Y···6BD12A···12P
order122222223···3446···66···612···12
size111122221···1221···12···22···2

90 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C3C6C6C6D4C3×D4
kernelD4×C3×C6C6×C12D4×C32C2×C62C6×D4C2×C12C3×D4C22×C6C3×C6C6
# reps1142883216216

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

300
090
009
,
1200
040
004
,
1200
01211
011
,
1200
01211
001
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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