Q8:3Dic9 - GroupNames

G = Q₈⋊₃Dic₉ order 288 = 2⁵·3²

2^nd semidirect product of Q₈ and Dic₉ acting via Dic₉/C₁₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈⋊₃Dic₉, D₄⋊₂Dic₉, C₃₆.₅₆D₄, C₉⋊₃C₄≀C₂, (D₄×C₉)⋊₂C₄, (Q₈×C₉)⋊₂C₄, C₃₆.₉(C₂×C₄), C₄○D₄.₃D₉, (C₂×C₁₈).₃D₄, (C₄×Dic₉)⋊₂C₂, (C₂×C₁₂).₄₉D₆, (C₂×C₄).₄₃D₁₈, C₄.Dic₉⋊₄C₂, C₄.₃(C₂×Dic₉), C₄.₃₁(C₉⋊D₄), (C₃×Q₈).₆Dic₃, (C₃×D₄).₂Dic₃, C₁₂.₃(C₂×Dic₃), C₃.(Q₈⋊₃Dic₃), (C₂×C₃₆).₂₇C₂², C₂².₃(C₉⋊D₄), C₁₂.₁₂₆(C₃⋊D₄), C₁₈.₁₈(C₂²⋊C₄), C₂.₈(C₁₈.D₄), C₆.₁₉(C₆.D₄), (C₃×C₄○D₄).₉S₃, (C₉×C₄○D₄).₁C₂, (C₂×C₆).₃(C₃⋊D₄), SmallGroup(288,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — Q₈⋊₃Dic₉

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₂×C₃₆ — C₄.Dic₉ — Q₈⋊₃Dic₉

Lower central

C₉ — C₁₈ — C₃₆ — Q₈⋊₃Dic₉

Upper central

C₁ — C₄ — C₂×C₄ — C₄○D₄

Generators and relations for Q₈⋊₃Dic₉
G = < a,b,c,d | a⁴=c¹⁸=1, b²=a², d²=c⁹, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=a²b, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 212 in 66 conjugacy classes, 30 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₉, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², M₄(2), C₄○D₄, C₁₈, C₁₈, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₄≀C₂, Dic₉, C₃₆, C₃₆, C₂×C₁₈, C₂×C₁₈, C₄.Dic₃, C₄×Dic₃, C₃×C₄○D₄, C₉⋊C₈, C₂×Dic₉, C₂×C₃₆, C₂×C₃₆, D₄×C₉, D₄×C₉, Q₈×C₉, Q₈⋊₃Dic₃, C₄.Dic₉, C₄×Dic₉, C₉×C₄○D₄, Q₈⋊₃Dic₉
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, Dic₃, D₆, C₂²⋊C₄, D₉, C₂×Dic₃, C₃⋊D₄, C₄≀C₂, Dic₉, D₁₈, C₆.D₄, C₂×Dic₉, C₉⋊D₄, Q₈⋊₃Dic₃, C₁₈.D₄, Q₈⋊₃Dic₉

Smallest permutation representation of Q₈⋊₃Dic₉
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	+	+	-	-	+				+	-	-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₆	Dic₃	Dic₃	D₉	C₃⋊D₄	C₃⋊D₄	C₄≀C₂	D₁₈	Dic₉	Dic₉	C₉⋊D₄	C₉⋊D₄	Q₈⋊₃Dic₃	Q₈⋊₃Dic₉
kernel	Q₈⋊₃Dic₉	C₄.Dic₉	C₄×Dic₉	C₉×C₄○D₄	D₄×C₉	Q₈×C₉	C₃×C₄○D₄	C₃₆	C₂×C₁₈	C₂×C₁₂	C₃×D₄	C₃×Q₈	C₄○D₄	C₁₂	C₂×C₆	C₉	C₂×C₄	D₄	Q₈	C₄	C₂²	C₃	C₁
# reps	1	1	1	1	2	2	1	1	1	1	1	1	3	2	2	4	3	3	3	6	6	2	6

Matrix representation of Q₈⋊₃Dic₉ ►in GL₄(𝔽₇₃) generated by

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

0	27	0	0
27	0	0	0
0	0	72	0
0	0	0	72

72	0	0	0
0	1	0	0
0	0	45	31
0	0	42	3

46	0	0	0
0	1	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(73))| [27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,27,0,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,45,42,0,0,31,3],[46,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

Q₈⋊₃Dic₉ in GAP, Magma, Sage, TeX

Q_8\rtimes_3{\rm Dic}_9

% in TeX

G:=Group("Q8:3Dic9");

// GroupNames label

G:=SmallGroup(288,44);

// by ID

G=gap.SmallGroup(288,44);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,100,675,346,80,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^18=1,b^2=a^2,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;

// generators/relations

G = Q8⋊3Dic9 order 288 = 25·32

2nd semidirect product of Q8 and Dic9 acting via Dic9/C18=C2

G = Q₈⋊₃Dic₉ order 288 = 2⁵·3²

2^nd semidirect product of Q₈ and Dic₉ acting via Dic₉/C₁₈=C₂