Q8:3D15 - GroupNames

G = Q₈:₃D₁₅ order 240 = 2⁴·3·5

The semidirect product of Q₈ and D₁₅ acting through Inn(Q₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₆₀:₄C₂, Q₈:₃D₁₅, C₄.₇D₃₀, C₂₀.₂₁D₆, C₁₂.₂₁D₁₀, C₆₀.₇C₂², C₃₀.₃₅C₂³, D₃₀.₇C₂², Dic₁₅.₁₇C₂², (C₅xQ₈):₅S₃, (C₃xQ₈):₃D₅, (C₄xD₁₅):₃C₂, (Q₈xC₁₅):₃C₂, C₁₅:₁₇(C₄oD₄), C₅:₃(Q₈:₃S₃), C₃:₃(Q₈:₂D₅), C₆.₃₅(C₂²xD₅), C₂.₉(C₂²xD₁₅), C₁₀.₃₅(C₂²xS₃), SmallGroup(240,182)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — Q₈:₃D₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — D₃₀ — C₄xD₁₅ — Q₈:₃D₁₅

Lower central

C₁₅ — C₃₀ — Q₈:₃D₁₅

Upper central

C₁ — C₂ — Q₈

Generators and relations for Q₈:₃D₁₅
G = < a,b,c,d | a⁴=c¹⁵=d²=1, b²=a², bab^-1=dad=a^-1, ac=ca, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 432 in 80 conjugacy classes, 35 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₂xC₄, D₄, Q₈, D₅, C₁₀, Dic₃, C₁₂, D₆, C₁₅, C₄oD₄, Dic₅, C₂₀, D₁₀, C₄xS₃, D₁₂, C₃xQ₈, D₁₅, C₃₀, C₄xD₅, D₂₀, C₅xQ₈, Q₈:₃S₃, Dic₁₅, C₆₀, D₃₀, Q₈:₂D₅, C₄xD₁₅, D₆₀, Q₈xC₁₅, Q₈:₃D₁₅
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄oD₄, D₁₀, C₂²xS₃, D₁₅, C₂²xD₅, Q₈:₃S₃, D₃₀, Q₈:₂D₅, C₂²xD₁₅, Q₈:₃D₁₅

Smallest permutation representation of Q₈:₃D₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

Q₈:₃D₁₅ is a maximal subgroup of
D₆₀:C₂² C₆₀.C₂³ D₂₀.₁₆D₆ D₁₂.D₁₀ Q₈:₃D₃₀ D₄.₅D₃₀ Q₁₆:D₁₅ D₁₂₀:₈C₂ C₃₀.₃₃C₂⁴ D₅xQ₈:₃S₃ S₃xQ₈:₂D₅ D₂₀:₁₇D₆ Q₈.₁₅D₃₀ C₄oD₄xD₁₅ D₄:₈D₃₀
Q₈:₃D₁₅ is a maximal quotient of
C₄.Dic₃₀ C₄:C₄:₇D₁₅ D₆₀:₁₁C₄ D₃₀.₂₉D₄ C₄:D₆₀ C₄:C₄:D₁₅ Q₈xDic₁₅ D₃₀:₇Q₈ C₆₀.₂₃D₄

45 conjugacy classes

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

45 irreducible representations

dim

type

image

C₁

C₂

S₃

D₅

D₆

C₄oD₄

D₁₀

D₁₅

D₃₀

Q₈:₃S₃

Q₈:₂D₅

Q₈:₃D₁₅

kernel

Q₈:₃D₁₅

C₄xD₁₅

D₆₀

Q₈xC₁₅

C₅xQ₈

C₃xQ₈

C₂₀

C₁₅

C₁₂

Q₈

C₄

C₅

C₃

C₁

# reps

Matrix representation of Q₈:₃D₁₅ ►in GL₄(F₆₁) generated by

60	0	0	0
0	60	0	0
0	0	0	1
0	0	60	0

1	0	0	0
0	1	0	0
0	0	0	50
0	0	50	0

9	56	0	0
5	38	0	0
0	0	1	0
0	0	0	1

52	5	0	0
45	9	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,50,0,0,50,0],[9,5,0,0,56,38,0,0,0,0,1,0,0,0,0,1],[52,45,0,0,5,9,0,0,0,0,0,1,0,0,1,0] >;

Q₈:₃D₁₅ in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_{15}

% in TeX

G:=Group("Q8:3D15");

// GroupNames label

G:=SmallGroup(240,182);

// by ID

G=gap.SmallGroup(240,182);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,964,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^15=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = Q8:3D15 order 240 = 24·3·5

The semidirect product of Q8 and D15 acting through Inn(Q8)

G = Q₈:₃D₁₅ order 240 = 2⁴·3·5

The semidirect product of Q₈ and D₁₅ acting through Inn(Q₈)