C9xC8:C2^2 - GroupNames

G = C₉×C₈⋊C₂² order 288 = 2⁵·3²

Direct product of C₉ and C₈⋊C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₉×C₈⋊C₂², D₈⋊₂C₁₈, C₇₂⋊₇C₂², C₃₆.₆₃D₄, SD₁₆⋊₁C₁₈, M₄(2)⋊₁C₁₈, C₃₆.₄₈C₂³, C₈⋊(C₂×C₁₈), (C₉×D₈)⋊₆C₂, C₄○D₄⋊₄C₁₈, D₄⋊₂(C₂×C₁₈), (C₂×D₄)⋊₅C₁₈, Q₈⋊₃(C₂×C₁₈), (C₃×D₈).₅C₆, C₄.₁₄(D₄×C₉), C₆.₇₈(C₆×D₄), (D₄×C₁₈)⋊₁₄C₂, C₂₄.₁₂(C₂×C₆), (C₉×SD₁₆)⋊₅C₂, (C₆×D₄).₁₆C₆, C₁₈.₇₈(C₂×D₄), C₂.₁₅(D₄×C₁₈), C₁₂.₇₃(C₃×D₄), (C₂×C₁₈).₂₅D₄, C₂².₅(D₄×C₉), (D₄×C₉)⋊₁₁C₂², (C₉×M₄(2))⋊₅C₂, C₄.₅(C₂²×C₁₈), (C₃×SD₁₆).₁C₆, (Q₈×C₉)⋊₁₀C₂², (C₂×C₃₆).₆₇C₂², C₁₂.₄₈(C₂²×C₆), (C₃×M₄(2)).₁C₆, C₃.(C₃×C₈⋊C₂²), (C₉×C₄○D₄)⋊₇C₂, (C₃×C₈⋊C₂²).C₃, (C₂×C₄).₄(C₂×C₁₈), (C₂×C₆).₂₉(C₃×D₄), (C₂×C₁₂).₆₅(C₂×C₆), (C₃×C₄○D₄).₁₄C₆, (C₃×D₄).₁₄(C₂×C₆), (C₃×Q₈).₂₇(C₂×C₆), SmallGroup(288,186)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₉×C₈⋊C₂²

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃₆ — D₄×C₉ — C₉×D₈ — C₉×C₈⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₉×C₈⋊C₂²

Upper central

C₁ — C₁₈ — C₂×C₃₆ — C₉×C₈⋊C₂²

Generators and relations for C₉×C₈⋊C₂²
G = < a,b,c,d | a⁹=b⁸=c²=d²=1, ab=ba, ac=ca, ad=da, cbc=b³, dbd=b⁵, cd=dc >

Subgroups: 174 in 102 conjugacy classes, 60 normal (36 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, D₄, Q₈, C₂³, C₉, C₁₂, C₁₂, C₂×C₆, C₂×C₆, M₄(2), D₈, SD₁₆, C₂×D₄, C₄○D₄, C₁₈, C₁₈, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×D₄, C₃×Q₈, C₂²×C₆, C₈⋊C₂², C₃₆, C₃₆, C₂×C₁₈, C₂×C₁₈, C₃×M₄(2), C₃×D₈, C₃×SD₁₆, C₆×D₄, C₃×C₄○D₄, C₇₂, C₂×C₃₆, C₂×C₃₆, D₄×C₉, D₄×C₉, D₄×C₉, Q₈×C₉, C₂²×C₁₈, C₃×C₈⋊C₂², C₉×M₄(2), C₉×D₈, C₉×SD₁₆, D₄×C₁₈, C₉×C₄○D₄, C₉×C₈⋊C₂²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₉, C₂×C₆, C₂×D₄, C₁₈, C₃×D₄, C₂²×C₆, C₈⋊C₂², C₂×C₁₈, C₆×D₄, D₄×C₉, C₂²×C₁₈, C₃×C₈⋊C₂², D₄×C₁₈, C₉×C₈⋊C₂²

Smallest permutation representation of C₉×C₈⋊C₂²
►On 72 points

Generators in S₇₂

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

(10 70)(11 71)(12 72)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 33)(20 34)(21 35)(22 36)(23 28)(24 29)(25 30)(26 31)(27 32)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)

(10 27)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 64)(36 65)

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

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

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

99 conjugacy classes

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

99 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+													+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₉	C₁₈	C₁₈	C₁₈	C₁₈	C₁₈	D₄	D₄	C₃×D₄	C₃×D₄	D₄×C₉	D₄×C₉	C₈⋊C₂²	C₃×C₈⋊C₂²	C₉×C₈⋊C₂²
kernel	C₉×C₈⋊C₂²	C₉×M₄(2)	C₉×D₈	C₉×SD₁₆	D₄×C₁₈	C₉×C₄○D₄	C₃×C₈⋊C₂²	C₃×M₄(2)	C₃×D₈	C₃×SD₁₆	C₆×D₄	C₃×C₄○D₄	C₈⋊C₂²	M₄(2)	D₈	SD₁₆	C₂×D₄	C₄○D₄	C₃₆	C₂×C₁₈	C₁₂	C₂×C₆	C₄	C₂²	C₉	C₃	C₁
# reps	1	1	2	2	1	1	2	2	4	4	2	2	6	6	12	12	6	6	1	1	2	2	6	6	1	2	6

Matrix representation of C₉×C₈⋊C₂² ►in GL₆(𝔽₇₃)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	8	0
0	0	0	0	0	8

0	72	0	0	0	0
1	0	0	0	0	0
0	0	17	0	71	0
0	0	17	0	72	72
0	0	71	1	56	0
0	0	71	0	56	0

72	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	1	72	0	0
0	0	17	0	0	72
0	0	17	0	72	0

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	17	0	72	0
0	0	17	0	0	72

G:=sub<GL(6,GF(73))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,17,17,71,71,0,0,0,0,1,0,0,0,71,72,56,56,0,0,0,72,0,0],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,17,17,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,17,17,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

C₉×C₈⋊C₂² in GAP, Magma, Sage, TeX

C_9\times C_8\rtimes C_2^2

% in TeX

G:=Group("C9xC8:C2^2");

// GroupNames label

G:=SmallGroup(288,186);

// by ID

G=gap.SmallGroup(288,186);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,3110,192,5884,2951,242]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;

// generators/relations

G = C9×C8⋊C22 order 288 = 25·32

Direct product of C9 and C8⋊C22

G = C₉×C₈⋊C₂² order 288 = 2⁵·3²

Direct product of C₉ and C₈⋊C₂²