C9xQ16 - GroupNames

G = C₉xQ₁₆ order 144 = 2⁴·3²

Direct product of C₉ and Q₁₆

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

Aliases: C₉xQ₁₆, C₈.C₁₈, C₂₄.₃C₆, C₇₂.₃C₂, C₁₈.₁₆D₄, Q₈.₂C₁₈, C₃₆.₁₉C₂², C₃.(C₃xQ₁₆), C₂.₅(D₄xC₉), (C₃xQ₁₆).C₃, C₄.₃(C₂xC₁₈), C₆.₁₆(C₃xD₄), (Q₈xC₉).₂C₂, (C₃xQ₈).₇C₆, C₁₂.₁₉(C₂xC₆), SmallGroup(144,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₉xQ₁₆

Chief series

C₁ — C₂ — C₆ — C₁₂ — C₃₆ — Q₈xC₉ — C₉xQ₁₆

Lower central

C₁ — C₂ — C₄ — C₉xQ₁₆

Upper central

C₁ — C₁₈ — C₃₆ — C₉xQ₁₆

Generators and relations for C₉xQ₁₆
G = < a,b,c | a⁹=b⁸=1, c²=b⁴, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 33 in 27 conjugacy classes, 21 normal (15 characteristic)
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₉, C₂xC₆, Q₁₆, C₁₈, C₃xD₄, C₂xC₁₈, C₃xQ₁₆, D₄xC₉, C₉xQ₁₆

C₁

C₂

C₃

C₄

₂C₄

C₆

C₉

C₈

Q₈

C₁₂

₂C₁₂

C₁₈

Q₁₆

C₂₄

C₃xQ₈

C₃₆

₂C₃₆

C₃xQ₁₆

Q₈xC₉

C₇₂

C₉xQ₁₆

Smallest permutation representation of C₉xQ₁₆
►Regular action on 144 points

Generators in S₁₄₄

(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)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)

(1 138 53 25 42 28 59 13)(2 139 54 26 43 29 60 14)(3 140 46 27 44 30 61 15)(4 141 47 19 45 31 62 16)(5 142 48 20 37 32 63 17)(6 143 49 21 38 33 55 18)(7 144 50 22 39 34 56 10)(8 136 51 23 40 35 57 11)(9 137 52 24 41 36 58 12)(64 112 87 100 93 123 76 129)(65 113 88 101 94 124 77 130)(66 114 89 102 95 125 78 131)(67 115 90 103 96 126 79 132)(68 116 82 104 97 118 80 133)(69 117 83 105 98 119 81 134)(70 109 84 106 99 120 73 135)(71 110 85 107 91 121 74 127)(72 111 86 108 92 122 75 128)

(1 78 42 89)(2 79 43 90)(3 80 44 82)(4 81 45 83)(5 73 37 84)(6 74 38 85)(7 75 39 86)(8 76 40 87)(9 77 41 88)(10 128 22 108)(11 129 23 100)(12 130 24 101)(13 131 25 102)(14 132 26 103)(15 133 27 104)(16 134 19 105)(17 135 20 106)(18 127 21 107)(28 114 138 125)(29 115 139 126)(30 116 140 118)(31 117 141 119)(32 109 142 120)(33 110 143 121)(34 111 144 122)(35 112 136 123)(36 113 137 124)(46 97 61 68)(47 98 62 69)(48 99 63 70)(49 91 55 71)(50 92 56 72)(51 93 57 64)(52 94 58 65)(53 95 59 66)(54 96 60 67)

G:=sub<Sym(144)| (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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,138,53,25,42,28,59,13)(2,139,54,26,43,29,60,14)(3,140,46,27,44,30,61,15)(4,141,47,19,45,31,62,16)(5,142,48,20,37,32,63,17)(6,143,49,21,38,33,55,18)(7,144,50,22,39,34,56,10)(8,136,51,23,40,35,57,11)(9,137,52,24,41,36,58,12)(64,112,87,100,93,123,76,129)(65,113,88,101,94,124,77,130)(66,114,89,102,95,125,78,131)(67,115,90,103,96,126,79,132)(68,116,82,104,97,118,80,133)(69,117,83,105,98,119,81,134)(70,109,84,106,99,120,73,135)(71,110,85,107,91,121,74,127)(72,111,86,108,92,122,75,128), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,128,22,108)(11,129,23,100)(12,130,24,101)(13,131,25,102)(14,132,26,103)(15,133,27,104)(16,134,19,105)(17,135,20,106)(18,127,21,107)(28,114,138,125)(29,115,139,126)(30,116,140,118)(31,117,141,119)(32,109,142,120)(33,110,143,121)(34,111,144,122)(35,112,136,123)(36,113,137,124)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67)>;

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)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,138,53,25,42,28,59,13)(2,139,54,26,43,29,60,14)(3,140,46,27,44,30,61,15)(4,141,47,19,45,31,62,16)(5,142,48,20,37,32,63,17)(6,143,49,21,38,33,55,18)(7,144,50,22,39,34,56,10)(8,136,51,23,40,35,57,11)(9,137,52,24,41,36,58,12)(64,112,87,100,93,123,76,129)(65,113,88,101,94,124,77,130)(66,114,89,102,95,125,78,131)(67,115,90,103,96,126,79,132)(68,116,82,104,97,118,80,133)(69,117,83,105,98,119,81,134)(70,109,84,106,99,120,73,135)(71,110,85,107,91,121,74,127)(72,111,86,108,92,122,75,128), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,128,22,108)(11,129,23,100)(12,130,24,101)(13,131,25,102)(14,132,26,103)(15,133,27,104)(16,134,19,105)(17,135,20,106)(18,127,21,107)(28,114,138,125)(29,115,139,126)(30,116,140,118)(31,117,141,119)(32,109,142,120)(33,110,143,121)(34,111,144,122)(35,112,136,123)(36,113,137,124)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67) );

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),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,138,53,25,42,28,59,13),(2,139,54,26,43,29,60,14),(3,140,46,27,44,30,61,15),(4,141,47,19,45,31,62,16),(5,142,48,20,37,32,63,17),(6,143,49,21,38,33,55,18),(7,144,50,22,39,34,56,10),(8,136,51,23,40,35,57,11),(9,137,52,24,41,36,58,12),(64,112,87,100,93,123,76,129),(65,113,88,101,94,124,77,130),(66,114,89,102,95,125,78,131),(67,115,90,103,96,126,79,132),(68,116,82,104,97,118,80,133),(69,117,83,105,98,119,81,134),(70,109,84,106,99,120,73,135),(71,110,85,107,91,121,74,127),(72,111,86,108,92,122,75,128)], [(1,78,42,89),(2,79,43,90),(3,80,44,82),(4,81,45,83),(5,73,37,84),(6,74,38,85),(7,75,39,86),(8,76,40,87),(9,77,41,88),(10,128,22,108),(11,129,23,100),(12,130,24,101),(13,131,25,102),(14,132,26,103),(15,133,27,104),(16,134,19,105),(17,135,20,106),(18,127,21,107),(28,114,138,125),(29,115,139,126),(30,116,140,118),(31,117,141,119),(32,109,142,120),(33,110,143,121),(34,111,144,122),(35,112,136,123),(36,113,137,124),(46,97,61,68),(47,98,62,69),(48,99,63,70),(49,91,55,71),(50,92,56,72),(51,93,57,64),(52,94,58,65),(53,95,59,66),(54,96,60,67)]])

C₉xQ₁₆ is a maximal subgroup of C₉:SD₃₂ C₉:Q₃₂ Q₁₆:D₉ D₇₂:₅C₂

63 conjugacy classes

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

63 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+							+	-
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	D₄	Q₁₆	C₃xD₄	C₃xQ₁₆	D₄xC₉	C₉xQ₁₆
kernel	C₉xQ₁₆	C₇₂	Q₈xC₉	C₃xQ₁₆	C₂₄	C₃xQ₈	Q₁₆	C₈	Q₈	C₁₈	C₉	C₆	C₃	C₂	C₁
# reps	1	1	2	2	2	4	6	6	12	1	2	2	4	6	12

Matrix representation of C₉xQ₁₆ ►in GL₂(F₇₃) generated by

55	0
0	55

0	41
16	41

5	17
50	68

G:=sub<GL(2,GF(73))| [55,0,0,55],[0,16,41,41],[5,50,17,68] >;

C₉xQ₁₆ in GAP, Magma, Sage, TeX

C_9\times Q_{16}

% in TeX

G:=Group("C9xQ16");

// GroupNames label

G:=SmallGroup(144,27);

// by ID

G=gap.SmallGroup(144,27);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,432,169,439,122,2019,1017,165]);

// Polycyclic

G:=Group<a,b,c|a^9=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of C₉xQ₁₆ in TeX

G = C9xQ16 order 144 = 24·32

Direct product of C9 and Q16

G = C₉xQ₁₆ order 144 = 2⁴·3²

Direct product of C₉ and Q₁₆