C9xC4oD4 - GroupNames

G = C₉xC₄oD₄ order 144 = 2⁴·3²

Direct product of C₉ and C₄oD₄

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

Aliases: C₉xC₄oD₄, D₄:₂C₁₈, Q₈:₃C₁₈, C₁₈.₁₃C₂³, C₃₆.₂₁C₂², C₃₆ o(D₄xC₉), C₃₆ o(Q₈xC₉), (C₂xC₃₆):₇C₂, (C₂xC₄):₃C₁₈, (D₄xC₉):₅C₂, (Q₈xC₉):₅C₂, C₄.₆(C₂xC₁₈), (C₃xD₄).₆C₆, C₂².(C₂xC₁₈), (C₂xC₁₂).₁₂C₆, C₁₂.₂₂(C₂xC₆), (C₃xQ₈).₁₀C₆, C₆.₁₃(C₂²xC₆), C₂.₃(C₂²xC₁₈), (C₂xC₁₈).₂C₂², C₃.(C₃xC₄oD₄), C₃₆ o(C₃xC₄oD₄), (C₂xC₆).₄(C₂xC₆), (C₃xC₄oD₄).₂C₃, SmallGroup(144,50)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₉xC₄oD₄

Chief series

C₁ — C₃ — C₆ — C₁₈ — C₂xC₁₈ — D₄xC₉ — C₉xC₄oD₄

Lower central

C₁ — C₂ — C₉xC₄oD₄

Upper central

C₁ — C₃₆ — C₉xC₄oD₄

Generators and relations for C₉xC₄oD₄
G = < a,b,c,d | a⁹=b⁴=d²=1, c²=b², ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b²c >

Subgroups: 69 in 60 conjugacy classes, 51 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₆, C₆, C₂xC₄, D₄, Q₈, C₉, C₁₂, C₁₂, C₂xC₆, C₄oD₄, C₁₈, C₁₈, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃₆, C₃₆, C₂xC₁₈, C₃xC₄oD₄, C₂xC₃₆, D₄xC₉, Q₈xC₉, C₉xC₄oD₄
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, C₉, C₂xC₆, C₄oD₄, C₁₈, C₂²xC₆, C₂xC₁₈, C₃xC₄oD₄, C₂²xC₁₈, C₉xC₄oD₄

Smallest permutation representation of C₉xC₄oD₄
►On 72 points

Generators in S₇₂

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

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

(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 64)(18 65)(46 60)(47 61)(48 62)(49 63)(50 55)(51 56)(52 57)(53 58)(54 59)

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

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

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

C₉xC₄oD₄ is a maximal subgroup of
Q₈:₃Dic₉ D₄.Dic₉ D₄.D₁₈ D₄:D₁₈ D₄.₉D₁₈ D₄:₈D₁₈ D₄.₁₀D₁₈ Q₈.C₅₄ C₃₆.A₄
C₉xC₄oD₄ is a maximal quotient of
D₄xC₃₆ Q₈xC₃₆

90 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	4C	4D	4E	6A	6B	6C	···	6H	9A	···	9F	12A	12B	12C	12D	12E	···	12J	18A	···	18F	18G	···	18X	36A	···	36L	36M	···	36AD
order	1	2	2	2	2	3	3	4	4	4	4	4	6	6	6	···	6	9	···	9	12	12	12	12	12	···	12	18	···	18	18	···	18	36	···	36	36	···	36
size	1	1	2	2	2	1	1	1	1	2	2	2	1	1	2	···	2	1	···	1	1	1	1	1	2	···	2	1	···	1	2	···	2	1	···	1	2	···	2

90 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₉	C₁₈	C₁₈	C₁₈	C₄oD₄	C₃xC₄oD₄	C₉xC₄oD₄
kernel	C₉xC₄oD₄	C₂xC₃₆	D₄xC₉	Q₈xC₉	C₃xC₄oD₄	C₂xC₁₂	C₃xD₄	C₃xQ₈	C₄oD₄	C₂xC₄	D₄	Q₈	C₉	C₃	C₁
# reps	1	3	3	1	2	6	6	2	6	18	18	6	2	4	12

Matrix representation of C₉xC₄oD₄ ►in GL₂(F₃₇) generated by

9	0
0	9

31	0
0	31

15	36
4	22

1	0
30	36

G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,31],[15,4,36,22],[1,30,0,36] >;

C₉xC₄oD₄ in GAP, Magma, Sage, TeX

C_9\times C_4\circ D_4

% in TeX

G:=Group("C9xC4oD4");

// GroupNames label

G:=SmallGroup(144,50);

// by ID

G=gap.SmallGroup(144,50);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,313,122,165]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;

// generators/relations

G = C9xC4oD4 order 144 = 24·32

Direct product of C9 and C4oD4

G = C₉xC₄oD₄ order 144 = 2⁴·3²

Direct product of C₉ and C₄oD₄