SD16:D9 - GroupNames

G = SD₁₆⋊D₉ order 288 = 2⁵·3²

2^nd semidirect product of SD₁₆ and D₉ acting via D₉/C₉=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₈.₂D₁₈, SD₁₆⋊₂D₉, D₄.₄D₁₈, D₁₈.₈D₄, C₂₄.₃₀D₆, Q₈.₆D₁₈, Dic₃₆⋊₆C₂, C₃₆.₆C₂³, C₇₂.₉C₂², Dic₉.₁₀D₄, Dic₁₈.₂C₂², (Q₈×D₉)⋊₂C₂, C₈⋊D₉⋊₂C₂, D₄⋊₂D₉.C₂, D₄.D₉⋊₄C₂, (C₃×D₄).₆D₆, C₉⋊Q₁₆⋊₁C₂, C₂.₂₀(D₄×D₉), C₆.₉₄(S₃×D₄), C₉⋊C₈.₁C₂², (C₉×SD₁₆)⋊₂C₂, C₁₈.₃₂(C₂×D₄), C₉⋊₂(C₈.C₂²), C₃.(D₄.D₆), (C₃×Q₈).₂₆D₆, C₄.₆(C₂²×D₉), (C₃×SD₁₆).₂S₃, (D₄×C₉).₄C₂², (C₄×D₉).₃C₂², (Q₈×C₉).₁C₂², C₁₂.₄₅(C₂²×S₃), SmallGroup(288,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — SD₁₆⋊D₉

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₄×D₉ — Q₈×D₉ — SD₁₆⋊D₉

Lower central

C₉ — C₁₈ — C₃₆ — SD₁₆⋊D₉

Upper central

C₁ — C₂ — C₄ — SD₁₆

Generators and relations for SD₁₆⋊D₉
G = < a,b,c,d | a⁸=b²=c⁹=d²=1, bab=a³, ac=ca, dad=a⁵, bc=cb, dbd=a⁴b, dcd=c^-1 >

Subgroups: 396 in 90 conjugacy classes, 34 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂×C₄, D₄, D₄, Q₈, Q₈, C₉, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, M₄(2), SD₁₆, SD₁₆, Q₁₆, C₂×Q₈, C₄○D₄, D₉, C₁₈, C₁₈, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₃×Q₈, C₈.C₂², Dic₉, Dic₉, C₃₆, C₃₆, D₁₈, C₂×C₁₈, C₈⋊S₃, Dic₁₂, D₄.S₃, C₃⋊Q₁₆, C₃×SD₁₆, D₄⋊₂S₃, S₃×Q₈, C₉⋊C₈, C₇₂, Dic₁₈, Dic₁₈, C₄×D₉, C₄×D₉, C₂×Dic₉, C₉⋊D₄, D₄×C₉, Q₈×C₉, D₄.D₆, Dic₃₆, C₈⋊D₉, D₄.D₉, C₉⋊Q₁₆, C₉×SD₁₆, D₄⋊₂D₉, Q₈×D₉, SD₁₆⋊D₉
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, D₉, C₂²×S₃, C₈.C₂², D₁₈, S₃×D₄, C₂²×D₉, D₄.D₆, D₄×D₉, SD₁₆⋊D₉

Smallest permutation representation of SD₁₆⋊D₉
►On 144 points

Generators in S₁₄₄

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₆	D₆	D₆	D₉	D₁₈	D₁₈	D₁₈	C₈.C₂²	S₃×D₄	D₄.D₆	D₄×D₉	SD₁₆⋊D₉
kernel	SD₁₆⋊D₉	Dic₃₆	C₈⋊D₉	D₄.D₉	C₉⋊Q₁₆	C₉×SD₁₆	D₄⋊₂D₉	Q₈×D₉	C₃×SD₁₆	Dic₉	D₁₈	C₂₄	C₃×D₄	C₃×Q₈	SD₁₆	C₈	D₄	Q₈	C₉	C₆	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	1	1	1	3	3	3	3	1	1	2	3	6

Matrix representation of SD₁₆⋊D₉ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	66	13
0	0	0	0	13	7
0	0	13	7	0	0
0	0	7	60	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

42	45	0	0	0	0
28	70	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

42	70	0	0	0	0
28	31	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	0	0	72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,13,7,0,0,0,0,7,60,0,0,66,13,0,0,0,0,13,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[42,28,0,0,0,0,45,70,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,28,0,0,0,0,70,31,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

SD₁₆⋊D₉ in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_9

% in TeX

G:=Group("SD16:D9");

// GroupNames label

G:=SmallGroup(288,125);

// by ID

G=gap.SmallGroup(288,125);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,346,185,80,6725,292,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;

// generators/relations

G = SD16⋊D9 order 288 = 25·32

2nd semidirect product of SD16 and D9 acting via D9/C9=C2

G = SD₁₆⋊D₉ order 288 = 2⁵·3²

2^nd semidirect product of SD₁₆ and D₉ acting via D₉/C₉=C₂