D4:6D18 - GroupNames

G = D₄:₆D₁₈ order 288 = 2⁵·3²

2^nd semidirect product of D₄ and D₁₈ acting through Inn(D₄)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄:₆D₁₈, C₂³:₃D₁₈, D₃₆:₈C₂², C₁₈.₇C₂⁴, C₉:₁2₊¹⁺⁴, C₃₆.₂₁C₂³, D₁₈.₃C₂³, Dic₁₈:₈C₂², Dic₉.₄C₂³, (D₄xD₉):₄C₂, (C₂xD₄):₇D₉, (C₂xC₄):₃D₁₈, (D₄xC₁₈):₇C₂, D₄:₂D₉:₄C₂, C₃.(D₄:₆D₆), (C₂xC₃₆):₃C₂², (C₆xD₄).₁₄S₃, (C₃xD₄).₃₇D₆, (C₄xD₉):₁C₂², (D₄xC₉):₇C₂², C₉:D₄:₃C₂², D₃₆:₅C₂:₅C₂, C₂.₈(C₂³xD₉), (C₂xC₁₂).₁₀₀D₆, (C₂xC₁₈).₂C₂³, C₆.₄₄(S₃xC₂³), C₄.₂₁(C₂²xD₉), (C₂²xC₆).₆₀D₆, C₁₂.₆₁(C₂²xS₃), (C₂²xC₁₈):₅C₂², (C₂xDic₉):₄C₂², (C₂²xD₉):₃C₂², C₂².₆(C₂²xD₉), (C₂xC₉:D₄):₁₁C₂, (C₂xC₆).₂₂₃(C₂²xS₃), SmallGroup(288,358)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₈ — D₄:₆D₁₈

Chief series

C₁ — C₃ — C₉ — C₁₈ — D₁₈ — C₂²xD₉ — D₄xD₉ — D₄:₆D₁₈

Lower central

C₉ — C₁₈ — D₄:₆D₁₈

Upper central

C₁ — C₂ — C₂xD₄

Generators and relations for D₄:₆D₁₈
G = < a,b,c,d | a⁴=b²=c¹⁸=d²=1, bab=cac^-1=a^-1, ad=da, cbc^-1=dbd=a²b, dcd=c^-1 >

Subgroups: 1080 in 249 conjugacy classes, 102 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₂³, C₉, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂xD₄, C₂xD₄, C₄oD₄, D₉, C₁₈, C₁₈, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, 2₊¹⁺⁴, Dic₉, C₃₆, D₁₈, D₁₈, C₂xC₁₈, C₂xC₁₈, C₂xC₁₈, C₄oD₁₂, S₃xD₄, D₄:₂S₃, C₂xC₃:D₄, C₆xD₄, Dic₁₈, C₄xD₉, D₃₆, C₂xDic₉, C₉:D₄, C₂xC₃₆, D₄xC₉, C₂²xD₉, C₂²xC₁₈, D₄:₆D₆, D₃₆:₅C₂, D₄xD₉, D₄:₂D₉, C₂xC₉:D₄, D₄xC₁₈, D₄:₆D₁₈
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, D₉, C₂²xS₃, 2₊¹⁺⁴, D₁₈, S₃xC₂³, C₂²xD₉, D₄:₆D₆, C₂³xD₉, D₄:₆D₁₈

Smallest permutation representation of D₄:₆D₁₈
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

57 conjugacy classes

class	1	2A	2B	···	2F	2G	2H	2I	2J	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	6G	9A	9B	9C	12A	12B	18A	···	18I	18J	···	18U	36A	···	36F
order	1	2	2	···	2	2	2	2	2	3	4	4	4	4	4	4	6	6	6	6	6	6	6	9	9	9	12	12	18	···	18	18	···	18	36	···	36
size	1	1	2	···	2	18	18	18	18	2	2	2	18	18	18	18	2	2	2	4	4	4	4	2	2	2	4	4	2	···	2	4	···	4	4	···	4

57 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

D₉

D₁₈

2₊¹⁺⁴

D₄:₆D₆

D₄:₆D₁₈

kernel

D₄:₆D₁₈

D₃₆:₅C₂

D₄xD₉

D₄:₂D₉

C₂xC₉:D₄

D₄xC₁₈

C₆xD₄

C₂xC₁₂

C₃xD₄

C₂²xC₆

C₂xD₄

C₂xC₄

D₄

C₂³

C₉

C₃

C₁

# reps

Matrix representation of D₄:₆D₁₈ ►in GL₆(F₃₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	30	23
0	0	0	0	14	7
0	0	7	14	0	0
0	0	23	30	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

6	17	0	0	0	0
20	26	0	0	0	0
0	0	0	1	0	0
0	0	36	36	0	0
0	0	0	0	0	36
0	0	0	0	1	1

6	17	0	0	0	0
11	31	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	36
0	0	0	0	36	0

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,23,0,0,0,0,14,30,0,0,30,14,0,0,0,0,23,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,20,0,0,0,0,17,26,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1],[6,11,0,0,0,0,17,31,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0] >;

D₄:₆D₁₈ in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{18}

% in TeX

G:=Group("D4:6D18");

// GroupNames label

G:=SmallGroup(288,358);

// by ID

G=gap.SmallGroup(288,358);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;

// generators/relations

G = D4:6D18 order 288 = 25·32

2nd semidirect product of D4 and D18 acting through Inn(D4)

G = D₄:₆D₁₈ order 288 = 2⁵·3²

2^nd semidirect product of D₄ and D₁₈ acting through Inn(D₄)