M4(2)xD9 - GroupNames

G = M₄(2)xD₉ order 288 = 2⁵·3²

Direct product of M₄(2) and D₉

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2)xD₉, C₈:₆D₁₈, C₇₂:₆C₂², C₂₄.₄₄D₆, C₃₆.₃₈C₂³, (C₈xD₉):₇C₂, C₈:D₉:₅C₂, C₉:C₈:₁₁C₂², (C₄xD₉).₁C₄, C₄.₁₅(C₄xD₉), C₉:₂(C₂xM₄(2)), C₃.(S₃xM₄(2)), C₁₂.₁₁(C₄xS₃), C₃₆.₁₂(C₂xC₄), D₁₈.₆(C₂xC₄), (C₂xC₁₂).₅₁D₆, (C₂xC₄).₄₆D₁₈, C₄.Dic₉:₅C₂, C₂².₇(C₄xD₉), (C₉xM₄(2)):₃C₂, (C₂xDic₉).₆C₄, Dic₉.₈(C₂xC₄), (C₂²xD₉).₄C₄, C₄.₃₈(C₂²xD₉), C₁₈.₁₅(C₂²xC₄), (C₂xC₃₆).₂₉C₂², (C₄xD₉).₁₈C₂², (C₃xM₄(2)).₃S₃, C₁₂.₁₉₉(C₂²xS₃), C₆.₅₄(S₃xC₂xC₄), (C₂xC₄xD₉).₃C₂, C₂.₁₆(C₂xC₄xD₉), (C₂xC₆).₈(C₄xS₃), (C₂xC₁₈).₅(C₂xC₄), SmallGroup(288,116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₈ — M₄(2)xD₉

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₄xD₉ — C₂xC₄xD₉ — M₄(2)xD₉

Lower central

C₉ — C₁₈ — M₄(2)xD₉

Upper central

C₁ — C₄ — M₄(2)

Generators and relations for M₄(2)xD₉
G = < a,b,c,d | a⁸=b²=c⁹=d²=1, bab=a⁵, ac=ca, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 384 in 102 conjugacy classes, 48 normal (34 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, C₂³, C₉, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₈, M₄(2), M₄(2), C₂²xC₄, D₉, D₉, C₁₈, C₁₈, C₃:C₈, C₂₄, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₂xM₄(2), Dic₉, C₃₆, D₁₈, D₁₈, C₂xC₁₈, S₃xC₈, C₈:S₃, C₄.Dic₃, C₃xM₄(2), S₃xC₂xC₄, C₉:C₈, C₇₂, C₄xD₉, C₂xDic₉, C₂xC₃₆, C₂²xD₉, S₃xM₄(2), C₈xD₉, C₈:D₉, C₄.Dic₉, C₉xM₄(2), C₂xC₄xD₉, M₄(2)xD₉
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₆, M₄(2), C₂²xC₄, D₉, C₄xS₃, C₂²xS₃, C₂xM₄(2), D₁₈, S₃xC₂xC₄, C₄xD₉, C₂²xD₉, S₃xM₄(2), C₂xC₄xD₉, M₄(2)xD₉

Smallest permutation representation of M₄(2)xD₉
►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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)>;

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	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	2	3	4	4	4	4	4	4	6	6	8	8	8	8	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	9	9	18	2	1	1	2	9	9	18	2	4	2	2	2	2	18	18	18	18	2	2	2	2	2	4	2	2	2	4	4	4	4	4	4	4	2	···	2	4	4	4	4	···	4

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+				+	+	+		+			+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₆	D₆	M₄(2)	D₉	C₄xS₃	C₄xS₃	D₁₈	D₁₈	C₄xD₉	C₄xD₉	S₃xM₄(2)	M₄(2)xD₉
kernel	M₄(2)xD₉	C₈xD₉	C₈:D₉	C₄.Dic₉	C₉xM₄(2)	C₂xC₄xD₉	C₄xD₉	C₂xDic₉	C₂²xD₉	C₃xM₄(2)	C₂₄	C₂xC₁₂	D₉	M₄(2)	C₁₂	C₂xC₆	C₈	C₂xC₄	C₄	C₂²	C₃	C₁
# reps	1	2	2	1	1	1	4	2	2	1	2	1	4	3	2	2	6	3	6	6	2	6

Matrix representation of M₄(2)xD₉ ►in GL₄(F₇₃) generated by

46	0	0	0
0	46	0	0
0	0	1	3
0	0	15	72

1	0	0	0
0	1	0	0
0	0	1	0
0	0	48	72

70	45	0	0
28	42	0	0
0	0	1	0
0	0	0	1

3	42	0	0
45	70	0	0
0	0	72	0
0	0	0	72

G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,1,15,0,0,3,72],[1,0,0,0,0,1,0,0,0,0,1,48,0,0,0,72],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[3,45,0,0,42,70,0,0,0,0,72,0,0,0,0,72] >;

M₄(2)xD₉ in GAP, Magma, Sage, TeX

M_4(2)\times D_9

% in TeX

G:=Group("M4(2)xD9");

// GroupNames label

G:=SmallGroup(288,116);

// by ID

G=gap.SmallGroup(288,116);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,6725,292,9414]);

// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = M4(2)xD9 order 288 = 25·32

Direct product of M4(2) and D9

G = M₄(2)xD₉ order 288 = 2⁵·3²

Direct product of M₄(2) and D₉