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, (C₅×Q₈)⋊₆D₆, Q₈⋊₆(S₃×D₅), C₃⋊₄(D₅×SD₁₆), C₅⋊₄(S₃×SD₁₆), C₅⋊₂C₈⋊₁₈D₆, (Q₈×D₁₅)⋊₁C₂, (C₃×Q₈)⋊₃D₁₀, C₆.₇₅(D₄×D₅), Q₈⋊₂S₃⋊₂D₅, C₁₀.₇₆(S₃×D₄), C₁₅⋊₁₅(C₂×SD₁₆), D₁₅⋊₂C₈⋊₅C₂, D₁₂.D₅⋊₅C₂, C₃₀.₁₉₁(C₂×D₄), C₆.D₂₀⋊₅C₂, C₂₀⋊D₆.₁C₂, (Q₈×C₁₅)⋊₅C₂², C₂₀.₂₉(C₂²×S₃), (C₄×D₁₅).₉C₂², C₁₂.₂₉(C₂²×D₅), (C₅×D₁₂).₁₀C₂², (C₃×D₂₀).₁₀C₂², C₂.₂₈(D₁₀⋊D₆), C₄.₂₉(C₂×S₃×D₅), (C₃×Q₈⋊D₅)⋊₃C₂, (C₅×C₃⋊C₈)⋊₁₃C₂², (C₅×Q₈⋊₂S₃)⋊₃C₂, (C₃×C₅⋊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 — C₃×D₂₀ — 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, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₁₅, C₂×C₈, SD₁₆, C₂×D₄, C₂×Q₈, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₃×D₄, C₃×Q₈, C₂²×S₃, C₅×S₃, C₃×D₅, D₁₅, C₃₀, C₂×SD₁₆, C₅⋊₂C₈, C₄₀, Dic₁₀, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₅×Q₈, C₂²×D₅, S₃×C₈, C₂₄⋊C₂, D₄.S₃, Q₈⋊₂S₃, C₃×SD₁₆, S₃×D₄, S₃×Q₈, Dic₁₅, Dic₁₅, C₆₀, C₆₀, S₃×D₅, C₆×D₅, S₃×C₁₀, D₃₀, C₈×D₅, C₄₀⋊C₂, D₄.D₅, Q₈⋊D₅, C₅×SD₁₆, D₄×D₅, Q₈×D₅, S₃×SD₁₆, C₅×C₃⋊C₈, C₃×C₅⋊₂C₈, C₁₅⋊D₄, C₃×D₂₀, C₅×D₁₂, Dic₃₀, Dic₃₀, C₄×D₁₅, C₄×D₁₅, Q₈×C₁₅, C₂×S₃×D₅, D₅×SD₁₆, D₁₅⋊₂C₈, C₆.D₂₀, D₁₂.D₅, C₃×Q₈⋊D₅, C₅×Q₈⋊₂S₃, C₂₀⋊D₆, Q₈×D₁₅, D₁₅⋊SD₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, SD₁₆, C₂×D₄, D₁₀, C₂²×S₃, C₂×SD₁₆, C₂²×D₅, S₃×D₄, S₃×D₅, D₄×D₅, S₃×SD₁₆, C₂×S₃×D₅, D₅×SD₁₆, 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	S₃×D₄	S₃×D₅	D₄×D₅	S₃×SD₁₆	C₂×S₃×D₅	D₅×SD₁₆	D₁₀⋊D₆	D₁₅⋊SD₁₆
kernel	D₁₅⋊SD₁₆	D₁₅⋊₂C₈	C₆.D₂₀	D₁₂.D₅	C₃×Q₈⋊D₅	C₅×Q₈⋊₂S₃	C₂₀⋊D₆	Q₈×D₁₅	Q₈⋊D₅	Dic₁₅	D₃₀	Q₈⋊₂S₃	C₅⋊₂C₈	D₂₀	C₅×Q₈	D₁₅	C₃⋊C₈	D₁₂	C₃×Q₈	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