Q8:C9:4C6 - GroupNames

G = Q₈:C₉:₄C₆ order 432 = 2⁴·3³

3^rd semidirect product of Q₈:C₉ and C₆ acting via C₆/C₂=C₃

Aliases: Q₈:C₉:₄C₆, C₁₂.₉(C₃xA₄), C₆.₂₅(C₆xA₄), (C₃xC₁₂).₂A₄, Q₈.C₁₈:₂C₃, C₄.(C₃².A₄), C₃².(C₄.A₄), C₄oD₄:₂3_-¹⁺², (Q₈xC₃²).₁₃C₆, Q₈:3_-¹⁺²:₂C₂, Q₈.₂(C₂x3_-¹⁺²), (C₃xC₆).₈(C₂xA₄), C₃.₄(C₃xC₄.A₄), (C₃xQ₈).₁₀(C₃xC₆), C₂.₃(C₂xC₃².A₄), (C₃xC₄oD₄).₄C₃², (C₃²xC₄oD₄).₂C₃, SmallGroup(432,338)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₃xQ₈ — Q₈:C₉:₄C₆

Chief series

C₁ — C₂ — Q₈ — C₃xQ₈ — Q₈xC₃² — Q₈:3_-¹⁺² — Q₈:C₉:₄C₆

Lower central

Q₈ — C₃xQ₈ — Q₈:C₉:₄C₆

Upper central

C₁ — C₁₂ — C₃xC₁₂

Generators and relations for Q₈:C₉:₄C₆
G = < a,b,c,d | a⁴=c⁹=d⁶=1, b²=a², bab^-1=dad^-1=a^-1, cac^-1=b, cbc^-1=ab, bd=db, dcd^-1=a^-1c⁷ >

Subgroups: 194 in 70 conjugacy classes, 23 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₂xC₄, D₄, Q₈, C₉, C₃², C₁₂, C₁₂, C₂xC₆, C₄oD₄, C₁₈, C₃xC₆, C₃xC₆, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃xQ₈, 3_-¹⁺², C₃₆, C₃xC₁₂, C₃xC₁₂, C₆², C₃xC₄oD₄, C₃xC₄oD₄, C₂x3_-¹⁺², Q₈:C₉, C₆xC₁₂, D₄xC₃², Q₈xC₃², C₄x3_-¹⁺², Q₈.C₁₈, C₃²xC₄oD₄, Q₈:3_-¹⁺², Q₈:C₉:₄C₆
Quotients: C₁, C₂, C₃, C₆, C₃², A₄, C₃xC₆, C₂xA₄, 3_-¹⁺², C₃xA₄, C₄.A₄, C₂x3_-¹⁺², C₆xA₄, C₃².A₄, C₃xC₄.A₄, C₂xC₃².A₄, Q₈:C₉:₄C₆

Smallest permutation representation of Q₈:C₉:₄C₆
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	2	2	3	3	3	3	3	3	3	3	6
type	+	+							+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₄.A₄	C₃xC₄.A₄	A₄	C₂xA₄	3_-¹⁺²	C₃xA₄	C₂x3_-¹⁺²	C₆xA₄	C₃².A₄	C₂xC₃².A₄	Q₈:C₉:₄C₆
kernel	Q₈:C₉:₄C₆	Q₈:3_-¹⁺²	Q₈.C₁₈	C₃²xC₄oD₄	Q₈:C₉	Q₈xC₃²	C₃²	C₃	C₃xC₁₂	C₃xC₆	C₄oD₄	C₁₂	Q₈	C₆	C₄	C₂	C₁
# reps	1	1	6	2	6	2	6	12	1	1	2	2	2	2	6	6	4

Matrix representation of Q₈:C₉:₄C₆ ►in GL₅(F₃₇)

6	0	0	0	0
6	31	0	0	0
0	0	36	0	0
0	0	0	36	0
0	0	0	0	1

36	2	0	0	0
36	1	0	0	0
0	0	1	0	0
0	0	0	36	0
0	0	0	0	36

31	5	0	0	0
16	5	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	10	0	0

14	9	0	0	0
14	23	0	0	0
0	0	36	0	0
0	0	0	26	0
0	0	0	0	10

G:=sub<GL(5,GF(37))| [6,6,0,0,0,0,31,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[36,36,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[31,16,0,0,0,5,5,0,0,0,0,0,0,0,10,0,0,1,0,0,0,0,0,1,0],[14,14,0,0,0,9,23,0,0,0,0,0,36,0,0,0,0,0,26,0,0,0,0,0,10] >;

Q₈:C₉:₄C₆ in GAP, Magma, Sage, TeX

Q_8\rtimes C_9\rtimes_4C_6

% in TeX

G:=Group("Q8:C9:4C6");

// GroupNames label

G:=SmallGroup(432,338);

// by ID

G=gap.SmallGroup(432,338);

# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,134,261,1901,172,3414,285,124]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^9=d^6=1,b^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=a^-1*c^7>;

// generators/relations

G = Q8:C9:4C6 order 432 = 24·33

3rd semidirect product of Q8:C9 and C6 acting via C6/C2=C3

G = Q₈:C₉:₄C₆ order 432 = 2⁴·3³

3^rd semidirect product of Q₈:C₉ and C₆ acting via C₆/C₂=C₃