D15:Q8 - GroupNames

G = D₁₅⋊Q₈ order 240 = 2⁴·3·5

The semidirect product of D₁₅ and Q₈ acting via Q₈/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₅⋊Q₈, Dic₆⋊₄D₅, C₂₀.₁₆D₆, Dic₁₀⋊₄S₃, C₁₂.₁₈D₁₀, C₃₀.₇C₂³, Dic₅.₄D₆, C₆₀.₂₈C₂², Dic₃.₃D₁₀, D₃₀.₁₀C₂², Dic₁₅.₁₂C₂², C₅⋊₁(S₃×Q₈), C₃⋊₁(Q₈×D₅), C₁₅⋊Q₈⋊₃C₂, C₁₅⋊₃(C₂×Q₈), C₄.₂₁(S₃×D₅), (C₅×Dic₆)⋊₆C₂, (C₄×D₁₅).₃C₂, C₆.₇(C₂²×D₅), (C₃×Dic₁₀)⋊₆C₂, D₃₀.C₂.₁C₂, C₁₀.₇(C₂²×S₃), (C₅×Dic₃).₃C₂², (C₃×Dic₅).₄C₂², C₂.₁₁(C₂×S₃×D₅), SmallGroup(240,131)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — D₁₅⋊Q₈

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃×Dic₅ — D₃₀.C₂ — D₁₅⋊Q₈

Lower central

C₁₅ — C₃₀ — D₁₅⋊Q₈

Upper central

C₁ — C₂ — C₄

Generators and relations for D₁₅⋊Q₈
G = < a,b,c,d | a¹⁵=b²=c⁴=1, d²=c², bab=a^-1, ac=ca, dad^-1=a¹¹, bc=cb, dbd^-1=a¹⁰b, dcd^-1=c^-1 >

Subgroups: 312 in 76 conjugacy classes, 34 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₂×C₄, Q₈, D₅, C₁₀, Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₁₅, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, D₁₀, Dic₆, Dic₆, C₄×S₃, C₃×Q₈, D₁₅, C₃₀, Dic₁₀, Dic₁₀, C₄×D₅, C₅×Q₈, S₃×Q₈, C₅×Dic₃, C₃×Dic₅, Dic₁₅, C₆₀, D₃₀, Q₈×D₅, D₃₀.C₂, C₁₅⋊Q₈, C₃×Dic₁₀, C₅×Dic₆, C₄×D₁₅, D₁₅⋊Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₅, D₆, C₂×Q₈, D₁₀, C₂²×S₃, C₂²×D₅, S₃×Q₈, S₃×D₅, Q₈×D₅, C₂×S₃×D₅, D₁₅⋊Q₈

Smallest permutation representation of D₁₅⋊Q₈
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 109)(92 108)(93 107)(94 106)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)

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

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

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

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

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

D₁₅⋊Q₈ is a maximal subgroup of
C₄₀⋊₁₄D₆ Dic₁₀.D₆ D₃₀.₃D₄ D₃₀.₄D₄ Dic₁₀⋊D₆ D₃₀.₉D₄ D₁₅⋊Q₁₆ C₆₀.C₂³ D₂₀.₃₈D₆ C₃₀.C₂⁴ D₂₀⋊₂₄D₆ C₁₅⋊2_-¹⁺⁴ D₃₀.C₂³ C₃₀.₃₃C₂⁴ S₃×Q₈×D₅
D₁₅⋊Q₈ is a maximal quotient of
Dic₁₅⋊₅Q₈ Dic₁₅⋊₁Q₈ Dic₁₅⋊Q₈ Dic₁₅⋊₆Q₈ Dic₁₅.Q₈ Dic₁₅.₂Q₈ Dic₁₅⋊₇Q₈ D₃₀⋊₈Q₈ Dic₁₅.₄Q₈ D₃₀⋊₉Q₈ Dic₁₅⋊₈Q₈ D₃₀⋊₁₀Q₈ D₃₀.Q₈ D₃₀⋊Q₈ D₃₀⋊₂Q₈ D₃₀⋊₃Q₈ D₃₀⋊₄Q₈ D₃₀.₂Q₈ C₂₀⋊Dic₆

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	-	+	+	+	+	+	-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₅	D₆	D₆	D₁₀	D₁₀	S₃×Q₈	S₃×D₅	Q₈×D₅	C₂×S₃×D₅	D₁₅⋊Q₈
kernel	D₁₅⋊Q₈	D₃₀.C₂	C₁₅⋊Q₈	C₃×Dic₁₀	C₅×Dic₆	C₄×D₁₅	Dic₁₀	D₁₅	Dic₆	Dic₅	C₂₀	Dic₃	C₁₂	C₅	C₄	C₃	C₂	C₁
# reps	1	2	2	1	1	1	1	2	2	2	1	4	2	1	2	2	2	4

Matrix representation of D₁₅⋊Q₈ ►in GL₆(𝔽₆₁)

60	17	0	0	0	0
44	44	0	0	0	0
0	0	1	46	0	0
0	0	49	59	0	0
0	0	0	0	1	0
0	0	0	0	0	1

60	17	0	0	0	0
0	1	0	0	0	0
0	0	1	46	0	0
0	0	0	60	0	0
0	0	0	0	60	0
0	0	0	0	0	60

1	0	0	0	0	0
0	1	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	60	12
0	0	0	0	10	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	49	60	0	0
0	0	0	0	1	34
0	0	0	0	43	60

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,46,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,10,0,0,0,0,12,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,49,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,34,60] >;

D₁₅⋊Q₈ in GAP, Magma, Sage, TeX

D_{15}\rtimes Q_8

% in TeX

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

// GroupNames label

G:=SmallGroup(240,131);

// by ID

G=gap.SmallGroup(240,131);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,218,116,50,490,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;

// generators/relations

G = D15⋊Q8 order 240 = 24·3·5

The semidirect product of D15 and Q8 acting via Q8/C4=C2

G = D₁₅⋊Q₈ order 240 = 2⁴·3·5

The semidirect product of D₁₅ and Q₈ acting via Q₈/C₄=C₂