C9xC4.D4 - GroupNames

G = C₉×C₄.D₄ order 288 = 2⁵·3²

Direct product of C₉ and C₄.D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₉×C₄.D₄, C₂³.C₃₆, C₃₆.₅₈D₄, M₄(2)⋊₃C₁₈, C₄.₉(D₄×C₉), (C₆×D₄).₇C₆, (D₄×C₁₈).₈C₂, (C₂×D₄).₂C₁₈, C₁₂.₆₇(C₃×D₄), (C₉×M₄(2))⋊₉C₂, (C₂²×C₆).₃C₁₂, (C₂²×C₁₈).₁C₄, C₂².₃(C₂×C₃₆), (C₂×C₃₆).₅₈C₂², (C₃×M₄(2)).₃C₆, C₁₈.₂₂(C₂²⋊C₄), C₃.(C₃×C₄.D₄), (C₂×C₄).₁(C₂×C₁₈), C₂.₄(C₉×C₂²⋊C₄), (C₃×C₄.D₄).C₃, (C₂×C₆).₂₄(C₂×C₁₂), (C₂×C₁₈).₂₀(C₂×C₄), (C₂×C₁₂).₆₀(C₂×C₆), C₆.₂₂(C₃×C₂²⋊C₄), SmallGroup(288,50)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₉×C₄.D₄

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₂×C₁₂ — C₂×C₃₆ — C₉×M₄(2) — C₉×C₄.D₄

Lower central

C₁ — C₂ — C₂² — C₉×C₄.D₄

Upper central

C₁ — C₁₈ — C₂×C₃₆ — C₉×C₄.D₄

Generators and relations for C₉×C₄.D₄
G = < a,b,c,d | a⁹=b⁴=1, c⁴=b², d²=b, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, bd=db, dcd^-1=b^-1c³ >

Subgroups: 126 in 69 conjugacy classes, 36 normal (18 characteristic)
C₁, C₂, C₂^[×3], C₃, C₄^[×2], C₂², C₂²^[×4], C₆, C₆^[×3], C₈^[×2], C₂×C₄, D₄^[×2], C₂³^[×2], C₉, C₁₂^[×2], C₂×C₆, C₂×C₆^[×4], M₄(2)^[×2], C₂×D₄, C₁₈, C₁₈^[×3], C₂₄^[×2], C₂×C₁₂, C₃×D₄^[×2], C₂²×C₆^[×2], C₄.D₄, C₃₆^[×2], C₂×C₁₈, C₂×C₁₈^[×4], C₃×M₄(2)^[×2], C₆×D₄, C₇₂^[×2], C₂×C₃₆, D₄×C₉^[×2], C₂²×C₁₈^[×2], C₃×C₄.D₄, C₉×M₄(2)^[×2], D₄×C₁₈, C₉×C₄.D₄
Quotients: C₁, C₂^[×3], C₃, C₄^[×2], C₂², C₆^[×3], C₂×C₄, D₄^[×2], C₉, C₁₂^[×2], C₂×C₆, C₂²⋊C₄, C₁₈^[×3], C₂×C₁₂, C₃×D₄^[×2], C₄.D₄, C₃₆^[×2], C₂×C₁₈, C₃×C₂²⋊C₄, C₂×C₃₆, D₄×C₉^[×2], C₃×C₄.D₄, C₉×C₂²⋊C₄, C₉×C₄.D₄

Smallest permutation representation of C₉×C₄.D₄
►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 59 46 40)(2 60 47 41)(3 61 48 42)(4 62 49 43)(5 63 50 44)(6 55 51 45)(7 56 52 37)(8 57 53 38)(9 58 54 39)(10 29 24 67)(11 30 25 68)(12 31 26 69)(13 32 27 70)(14 33 19 71)(15 34 20 72)(16 35 21 64)(17 36 22 65)(18 28 23 66)

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

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

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

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

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

99 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	8C	8D	9A	···	9F	12A	12B	12C	12D	18A	···	18F	18G	···	18L	18M	···	18X	24A	···	24H	36A	···	36L	72A	···	72X
order	1	2	2	2	2	3	3	4	4	6	6	6	6	6	6	6	6	8	8	8	8	9	···	9	12	12	12	12	18	···	18	18	···	18	18	···	18	24	···	24	36	···	36	72	···	72
size	1	1	2	4	4	1	1	2	2	1	1	2	2	4	4	4	4	4	4	4	4	1	···	1	2	2	2	2	1	···	1	2	···	2	4	···	4	4	···	4	2	···	2	4	···	4

99 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	4	4	4
type	+	+	+										+			+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₉	C₁₂	C₁₈	C₁₈	C₃₆	D₄	C₃×D₄	D₄×C₉	C₄.D₄	C₃×C₄.D₄	C₉×C₄.D₄
kernel	C₉×C₄.D₄	C₉×M₄(2)	D₄×C₁₈	C₃×C₄.D₄	C₂²×C₁₈	C₃×M₄(2)	C₆×D₄	C₄.D₄	C₂²×C₆	M₄(2)	C₂×D₄	C₂³	C₃₆	C₁₂	C₄	C₉	C₃	C₁
# reps	1	2	1	2	4	4	2	6	8	12	6	24	2	4	12	1	2	6

Matrix representation of C₉×C₄.D₄ ►in GL₆(𝔽₇₃)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	71	0	0
0	0	1	1	0	0
0	0	0	46	0	1
0	0	27	27	72	0

0	1	0	0	0	0
72	0	0	0	0	0
0	0	27	0	71	0
0	0	46	0	1	72
0	0	0	1	46	0
0	0	0	0	27	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	27	0	71	0
0	0	0	0	1	1
0	0	0	1	46	0
0	0	1	0	27	0

G:=sub<GL(6,GF(73))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,27,0,0,71,1,46,27,0,0,0,0,0,72,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,72,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,1,0,0] >;

C₉×C₄.D₄ in GAP, Magma, Sage, TeX

C_9\times C_4.D_4

% in TeX

G:=Group("C9xC4.D4");

// GroupNames label

G:=SmallGroup(288,50);

// by ID

G=gap.SmallGroup(288,50);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2951,242]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;

// generators/relations

G = C9×C4.D4 order 288 = 25·32

Direct product of C9 and C4.D4

G = C₉×C₄.D₄ order 288 = 2⁵·3²

Direct product of C₉ and C₄.D₄