D15:SD16 - GroupNames

G = D₁₅:SD₁₆ order 480 = 2⁵·3·5

The semidirect product of D₁₅ and SD₁₆ acting via SD₁₆/Q₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₅:₃SD₁₆, D₂₀.₁₁D₆, D₃₀.₄₁D₄, D₁₂.₁₁D₁₀, C₆₀.₂₉C₂³, Dic₁₅.₁₅D₄, Dic₃₀:₆C₂², C₃:C₈:₁₈D₁₀, Q₈:D₅:₂S₃, (C₅xQ₈):₆D₆, Q₈:₆(S₃xD₅), C₃:₄(D₅xSD₁₆), C₅:₄(S₃xSD₁₆), C₅:₂C₈:₁₈D₆, (Q₈xD₁₅):₁C₂, (C₃xQ₈):₃D₁₀, C₆.₇₅(D₄xD₅), Q₈:₂S₃:₂D₅, C₁₀.₇₆(S₃xD₄), C₁₅:₁₅(C₂xSD₁₆), D₁₅:₂C₈:₅C₂, D₁₂.D₅:₅C₂, C₃₀.₁₉₁(C₂xD₄), C₆.D₂₀:₅C₂, C₂₀:D₆.₁C₂, (Q₈xC₁₅):₅C₂², C₂₀.₂₉(C₂²xS₃), (C₄xD₁₅).₉C₂², C₁₂.₂₉(C₂²xD₅), (C₅xD₁₂).₁₀C₂², (C₃xD₂₀).₁₀C₂², C₂.₂₈(D₁₀:D₆), C₄.₂₉(C₂xS₃xD₅), (C₃xQ₈:D₅):₃C₂, (C₅xC₃:C₈):₁₃C₂², (C₅xQ₈:₂S₃):₃C₂, (C₃xC₅:₂C₈):₁₃C₂², SmallGroup(480,581)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — D₁₅:SD₁₆

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — C₃xD₂₀ — C₂₀:D₆ — D₁₅:SD₁₆

Lower central

C₁₅ — C₃₀ — C₆₀ — D₁₅:SD₁₆

Upper central

C₁ — C₂ — C₄ — Q₈

Generators and relations for D₁₅:SD₁₆
G = < a,b,c,d | a¹⁵=b²=c⁸=d²=1, bab=a^-1, cac^-1=dad=a¹¹, cbc^-1=dbd=a¹⁰b, dcd=c³ >

Subgroups: 924 in 136 conjugacy classes, 40 normal (38 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₆, C₈, C₂xC₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₁₅, C₂xC₈, SD₁₆, C₂xD₄, C₂xQ₈, Dic₅, C₂₀, C₂₀, D₁₀, C₂xC₁₀, C₃:C₈, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₃xD₄, C₃xQ₈, C₂²xS₃, C₅xS₃, C₃xD₅, D₁₅, C₃₀, C₂xSD₁₆, C₅:₂C₈, C₄₀, Dic₁₀, C₄xD₅, D₂₀, C₅:D₄, C₅xD₄, C₅xQ₈, C₂²xD₅, S₃xC₈, C₂₄:C₂, D₄.S₃, Q₈:₂S₃, C₃xSD₁₆, S₃xD₄, S₃xQ₈, Dic₁₅, Dic₁₅, C₆₀, C₆₀, S₃xD₅, C₆xD₅, S₃xC₁₀, D₃₀, C₈xD₅, C₄₀:C₂, D₄.D₅, Q₈:D₅, C₅xSD₁₆, D₄xD₅, Q₈xD₅, S₃xSD₁₆, C₅xC₃:C₈, C₃xC₅:₂C₈, C₁₅:D₄, C₃xD₂₀, C₅xD₁₂, Dic₃₀, Dic₃₀, C₄xD₁₅, C₄xD₁₅, Q₈xC₁₅, C₂xS₃xD₅, D₅xSD₁₆, D₁₅:₂C₈, C₆.D₂₀, D₁₂.D₅, C₃xQ₈:D₅, C₅xQ₈:₂S₃, C₂₀:D₆, Q₈xD₁₅, D₁₅:SD₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, SD₁₆, C₂xD₄, D₁₀, C₂²xS₃, C₂xSD₁₆, C₂²xD₅, S₃xD₄, S₃xD₅, D₄xD₅, S₃xSD₁₆, C₂xS₃xD₅, D₅xSD₁₆, D₁₀:D₆, D₁₅:SD₁₆

Smallest permutation representation of D₁₅:SD₁₆
►On 120 points

Generators in S₁₂₀

(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)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 99)(92 98)(93 97)(94 96)(100 105)(101 104)(102 103)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)

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

(2 12)(3 8)(5 15)(6 11)(9 14)(17 27)(18 23)(20 30)(21 26)(24 29)(31 57)(32 53)(33 49)(34 60)(35 56)(36 52)(37 48)(38 59)(39 55)(40 51)(41 47)(42 58)(43 54)(44 50)(45 46)(61 103)(62 99)(63 95)(64 91)(65 102)(66 98)(67 94)(68 105)(69 101)(70 97)(71 93)(72 104)(73 100)(74 96)(75 92)(76 118)(77 114)(78 110)(79 106)(80 117)(81 113)(82 109)(83 120)(84 116)(85 112)(86 108)(87 119)(88 115)(89 111)(90 107)

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

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

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

45 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	5A	5B	6A	6B	8A	8B	8C	8D	10A	10B	10C	10D	12A	12B	15A	15B	20A	20B	20C	20D	24A	24B	30A	30B	40A	40B	40C	40D	60A	···	60F
order	1	2	2	2	2	2	3	4	4	4	4	5	5	6	6	8	8	8	8	10	10	10	10	12	12	15	15	20	20	20	20	24	24	30	30	40	40	40	40	60	···	60
size	1	1	12	15	15	20	2	2	4	30	60	2	2	2	40	6	6	10	10	2	2	24	24	4	8	4	4	4	4	8	8	20	20	4	4	12	12	12	12	8	···	8

45 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+	+	+		+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₅	D₆	D₆	D₆	SD₁₆	D₁₀	D₁₀	D₁₀	S₃xD₄	S₃xD₅	D₄xD₅	S₃xSD₁₆	C₂xS₃xD₅	D₅xSD₁₆	D₁₀:D₆	D₁₅:SD₁₆
kernel	D₁₅:SD₁₆	D₁₅:₂C₈	C₆.D₂₀	D₁₂.D₅	C₃xQ₈:D₅	C₅xQ₈:₂S₃	C₂₀:D₆	Q₈xD₁₅	Q₈:D₅	Dic₁₅	D₃₀	Q₈:₂S₃	C₅:₂C₈	D₂₀	C₅xQ₈	D₁₅	C₃:C₈	D₁₂	C₃xQ₈	C₁₀	Q₈	C₆	C₅	C₄	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	4	2	2	2	1	2	2	2	2	4	4	2

Matrix representation of D₁₅:SD₁₆ ►in GL₆(F₂₄₁)

240	189	0	0	0	0
52	52	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	240	1
0	0	0	0	240	0

240	189	0	0	0	0
0	1	0	0	0	0
0	0	240	0	0	0
0	0	0	240	0	0
0	0	0	0	240	0
0	0	0	0	240	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	38	73	0	0
0	0	208	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	240	123	0	0
0	0	0	1	0	0
0	0	0	0	0	240
0	0	0	0	240	0

G:=sub<GL(6,GF(241))| [240,52,0,0,0,0,189,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,0,0,0,0,0,189,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,240,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,38,208,0,0,0,0,73,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,123,1,0,0,0,0,0,0,0,240,0,0,0,0,240,0] >;

D₁₅:SD₁₆ in GAP, Magma, Sage, TeX

D_{15}\rtimes {\rm SD}_{16}

% in TeX

G:=Group("D15:SD16");

// GroupNames label

G:=SmallGroup(480,581);

// by ID

G=gap.SmallGroup(480,581);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,100,675,346,185,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^3>;

// generators/relations

G = D15:SD16 order 480 = 25·3·5

The semidirect product of D15 and SD16 acting via SD16/Q8=C2

G = D₁₅:SD₁₆ order 480 = 2⁵·3·5

The semidirect product of D₁₅ and SD₁₆ acting via SD₁₆/Q₈=C₂