M4(2):D9 - GroupNames

G = M₄(2)⋊D₉ order 288 = 2⁵·3²

3^rd semidirect product of M₄(2) and D₉ acting via D₉/C₉=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂.₄D₁₂, C₄.₁₂D₃₆, C₃₆.₄₈D₄, M₄(2)⋊₃D₉, (C₂×C₄).₂D₁₈, (C₂²×D₉).C₄, (C₂×D₃₆).₆C₂, (C₂×C₁₂).₄₃D₆, C₉⋊₁(C₄.D₄), C₄.Dic₉⋊₂C₂, C₂².₅(C₄×D₉), C₆.₁₆(D₆⋊C₄), C₄.₂₂(C₉⋊D₄), (C₉×M₄(2))⋊₇C₂, C₂.₁₀(D₁₈⋊C₄), C₁₈.₉(C₂²⋊C₄), (C₂×C₃₆).₂₁C₂², C₃.(M₄(2)⋊S₃), (C₃×M₄(2)).₇S₃, C₁₂.₁₀₉(C₃⋊D₄), (C₂×C₆).₄(C₄×S₃), (C₂×C₁₈).₃(C₂×C₄), SmallGroup(288,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₈ — M₄(2)⋊D₉

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₂×C₃₆ — C₂×D₃₆ — M₄(2)⋊D₉

Lower central

C₉ — C₁₈ — C₂×C₁₈ — M₄(2)⋊D₉

Upper central

C₁ — C₂ — C₂×C₄ — M₄(2)

Generators and relations for M₄(2)⋊D₉
G = < a,b,c,d | a⁸=b²=c⁹=d²=1, bab=a⁵, ac=ca, dad=ab, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 436 in 69 conjugacy classes, 26 normal (24 characteristic)
C₁, C₂, C₂^[×3], C₃, C₄^[×2], C₂², C₂²^[×4], S₃^[×2], C₆, C₆, C₈^[×2], C₂×C₄, D₄^[×2], C₂³^[×2], C₉, C₁₂^[×2], D₆^[×4], C₂×C₆, M₄(2), M₄(2), C₂×D₄, D₉^[×2], C₁₈, C₁₈, C₃⋊C₈, C₂₄, D₁₂^[×2], C₂×C₁₂, C₂²×S₃^[×2], C₄.D₄, C₃₆^[×2], D₁₈^[×4], C₂×C₁₈, C₄.Dic₃, C₃×M₄(2), C₂×D₁₂, C₉⋊C₈, C₇₂, D₃₆^[×2], C₂×C₃₆, C₂²×D₉^[×2], M₄(2)⋊S₃, C₄.Dic₉, C₉×M₄(2), C₂×D₃₆, M₄(2)⋊D₉
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², S₃, C₂×C₄, D₄^[×2], D₆, C₂²⋊C₄, D₉, C₄×S₃, D₁₂, C₃⋊D₄, C₄.D₄, D₁₈, D₆⋊C₄, C₄×D₉, D₃₆, C₉⋊D₄, M₄(2)⋊S₃, D₁₈⋊C₄, M₄(2)⋊D₉

Smallest permutation representation of M₄(2)⋊D₉
►On 72 points

Generators in S₇₂

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

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

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

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

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

51 conjugacy classes

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

51 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+		+	+	+	+	+			+	+			+	+	+
image	C₁	C₂	C₂	C₂	C₄	S₃	D₄	D₆	D₉	D₁₂	C₃⋊D₄	C₄×S₃	D₁₈	D₃₆	C₉⋊D₄	C₄×D₉	C₄.D₄	M₄(2)⋊S₃	M₄(2)⋊D₉
kernel	M₄(2)⋊D₉	C₄.Dic₉	C₉×M₄(2)	C₂×D₃₆	C₂²×D₉	C₃×M₄(2)	C₃₆	C₂×C₁₂	M₄(2)	C₁₂	C₁₂	C₂×C₆	C₂×C₄	C₄	C₄	C₂²	C₉	C₃	C₁
# reps	1	1	1	1	4	1	2	1	3	2	2	2	3	6	6	6	1	2	6

Matrix representation of M₄(2)⋊D₉ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	71	0
0	0	0	72	0	71
0	0	4	66	1	0
0	0	7	70	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	72	0	72	0
0	0	0	72	0	72

16	0	0	0	0	0
0	32	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	72	1
0	0	0	0	72	0

0	32	0	0	0	0
16	0	0	0	0	0
0	0	66	14	0	0
0	0	7	7	0	0
0	0	0	0	66	14
0	0	0	0	7	7

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,4,7,0,0,0,72,66,70,0,0,71,0,1,0,0,0,0,71,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,72,0,0,0,0,72,0,0,0,0,0,0,72],[16,0,0,0,0,0,0,32,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,16,0,0,0,0,32,0,0,0,0,0,0,0,66,7,0,0,0,0,14,7,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;

M₄(2)⋊D₉ in GAP, Magma, Sage, TeX

M_{4(2})\rtimes D_9

% in TeX

G:=Group("M4(2):D9");

// GroupNames label

G:=SmallGroup(288,31);

// by ID

G=gap.SmallGroup(288,31);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = M4(2)⋊D9 order 288 = 25·32

3rd semidirect product of M4(2) and D9 acting via D9/C9=C2

G = M₄(2)⋊D₉ order 288 = 2⁵·3²

3^rd semidirect product of M₄(2) and D₉ acting via D₉/C₉=C₂