C8xC9:C6 - GroupNames

G = C₈×C₉⋊C₆ order 432 = 2⁴·3³

Direct product of C₈ and C₉⋊C₆

direct product, metacyclic, supersoluble, monomial

Aliases: C₈×C₉⋊C₆, D₉⋊C₂₄, C₇₂⋊₃C₆, D₁₈.₂C₁₂, Dic₉.₂C₁₂, C₉⋊C₈⋊₆C₆, (C₈×D₉)⋊C₃, C₉⋊C₂₄⋊₆C₂, C₉⋊₁(C₂×C₂₄), C₃².(S₃×C₈), C₉⋊C₁₂.₂C₄, C₃.₃(S₃×C₂₄), C₆.₉(S₃×C₁₂), (C₃×C₂₄).₉S₃, (C₄×D₉).₃C₆, C₂₄.₂₁(C₃×S₃), C₁₂.₈₈(S₃×C₆), C₃₆.₁₃(C₂×C₆), C₁₈.₁(C₂×C₁₂), (C₃×C₁₂).₅₈D₆, 3_-¹⁺²⋊₁(C₂×C₈), (C₈×3_-¹⁺²)⋊₃C₂, (C₄×3_-¹⁺²).₁₂C₂², C₂.₁(C₄×C₉⋊C₆), (C₂×C₉⋊C₆).₂C₄, (C₄×C₉⋊C₆).₃C₂, C₄.₁₂(C₂×C₉⋊C₆), (C₃×C₆).₁₂(C₄×S₃), (C₂×3_-¹⁺²).₁(C₂×C₄), SmallGroup(432,120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₉ — C₈×C₉⋊C₆

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₄×3_-¹⁺² — C₄×C₉⋊C₆ — C₈×C₉⋊C₆

Lower central

C₉ — C₈×C₉⋊C₆

Upper central

C₁ — C₈

Generators and relations for C₈×C₉⋊C₆
G = < a,b,c | a⁸=b⁹=c⁶=1, ab=ba, ac=ca, cbc^-1=b² >

Subgroups: 222 in 70 conjugacy classes, 34 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, C₉, C₉, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₈, D₉, C₁₈, C₁₈, C₃×S₃, C₃×C₆, C₃⋊C₈, C₂₄, C₂₄, C₄×S₃, C₂×C₁₂, 3_-¹⁺², Dic₉, C₃₆, C₃₆, D₁₈, C₃×Dic₃, C₃×C₁₂, S₃×C₆, S₃×C₈, C₂×C₂₄, C₉⋊C₆, C₂×3_-¹⁺², C₉⋊C₈, C₇₂, C₇₂, C₄×D₉, C₃×C₃⋊C₈, C₃×C₂₄, S₃×C₁₂, C₉⋊C₁₂, C₄×3_-¹⁺², C₂×C₉⋊C₆, C₈×D₉, S₃×C₂₄, C₉⋊C₂₄, C₈×3_-¹⁺², C₄×C₉⋊C₆, C₈×C₉⋊C₆
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₈, C₂×C₄, C₁₂, D₆, C₂×C₆, C₂×C₈, C₃×S₃, C₂₄, C₄×S₃, C₂×C₁₂, S₃×C₆, S₃×C₈, C₂×C₂₄, C₉⋊C₆, S₃×C₁₂, C₂×C₉⋊C₆, S₃×C₂₄, C₄×C₉⋊C₆, C₈×C₉⋊C₆

Smallest permutation representation of C₈×C₉⋊C₆
►On 72 points

Generators in S₇₂

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

(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)

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

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

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

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

80 conjugacy classes

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

80 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	6	6	6	6
type	+	+	+	+											+	+							+	+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₈	C₁₂	C₁₂	C₂₄	S₃	D₆	C₃×S₃	C₄×S₃	S₃×C₆	S₃×C₈	S₃×C₁₂	S₃×C₂₄	C₉⋊C₆	C₂×C₉⋊C₆	C₄×C₉⋊C₆	C₈×C₉⋊C₆
kernel	C₈×C₉⋊C₆	C₉⋊C₂₄	C₈×3_-¹⁺²	C₄×C₉⋊C₆	C₈×D₉	C₉⋊C₁₂	C₂×C₉⋊C₆	C₉⋊C₈	C₇₂	C₄×D₉	C₉⋊C₆	Dic₉	D₁₈	D₉	C₃×C₂₄	C₃×C₁₂	C₂₄	C₃×C₆	C₁₂	C₃²	C₆	C₃	C₈	C₄	C₂	C₁
# reps	1	1	1	1	2	2	2	2	2	2	8	4	4	16	1	1	2	2	2	4	4	8	1	1	2	4

Matrix representation of C₈×C₉⋊C₆ ►in GL₆(𝔽₇₃)

22	0	0	0	0	0
0	22	0	0	0	0
0	0	22	0	0	0
0	0	0	22	0	0
0	0	0	0	22	0
0	0	0	0	0	22

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

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

G:=sub<GL(6,GF(73))| [22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,22],[0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,1,72,0,0,0,0,0,72,0,0] >;

C₈×C₉⋊C₆ in GAP, Magma, Sage, TeX

C_8\times C_9\rtimes C_6

% in TeX

G:=Group("C8xC9:C6");

// GroupNames label

G:=SmallGroup(432,120);

// by ID

G=gap.SmallGroup(432,120);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,10085,2035,292,14118]);

// Polycyclic

G:=Group<a,b,c|a^8=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;

// generators/relations

G = C8×C9⋊C6 order 432 = 24·33

Direct product of C8 and C9⋊C6

G = C₈×C₉⋊C₆ order 432 = 2⁴·3³

Direct product of C₈ and C₉⋊C₆