Q8xD15 - GroupNames

G = Q₈×D₁₅ order 240 = 2⁴·3·5

Direct product of Q₈ and D₁₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈×D₁₅, C₄.₆D₃₀, C₂₀.₂₀D₆, Dic₃₀⋊₄C₂, C₁₂.₂₀D₁₀, C₆₀.₆C₂², C₃₀.₃₄C₂³, D₃₀.₁₆C₂², Dic₁₅.₉C₂², C₅⋊₃(S₃×Q₈), C₃⋊₃(Q₈×D₅), C₁₅⋊₈(C₂×Q₈), (C₃×Q₈)⋊₂D₅, (C₅×Q₈)⋊₄S₃, (Q₈×C₁₅)⋊₂C₂, (C₄×D₁₅).₁C₂, C₆.₃₄(C₂²×D₅), C₂.₈(C₂²×D₁₅), C₁₀.₃₄(C₂²×S₃), SmallGroup(240,181)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — Q₈×D₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — D₃₀ — C₄×D₁₅ — Q₈×D₁₅

Lower central

C₁₅ — C₃₀ — Q₈×D₁₅

Upper central

C₁ — C₂ — Q₈

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

Subgroups: 336 in 76 conjugacy classes, 37 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₂×C₄, Q₈, Q₈, D₅, C₁₀, Dic₃, C₁₂, D₆, C₁₅, C₂×Q₈, Dic₅, C₂₀, D₁₀, Dic₆, C₄×S₃, C₃×Q₈, D₁₅, C₃₀, Dic₁₀, C₄×D₅, C₅×Q₈, S₃×Q₈, Dic₁₅, C₆₀, D₃₀, Q₈×D₅, Dic₃₀, C₄×D₁₅, Q₈×C₁₅, Q₈×D₁₅
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₅, D₆, C₂×Q₈, D₁₀, C₂²×S₃, D₁₅, C₂²×D₅, S₃×Q₈, D₃₀, Q₈×D₅, C₂²×D₁₅, Q₈×D₁₅

Smallest permutation representation of Q₈×D₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

Q₈×D₁₅ is a maximal subgroup of
D₁₅⋊SD₁₆ D₁₅⋊Q₁₆ D₂₀.₁₇D₆ D₃₀.₄₄D₄ SD₁₆⋊D₁₅ Q₁₆⋊D₁₅ D₂₀.₂₉D₆ D₁₂.₂₉D₁₀ S₃×Q₈×D₅ D₂₀⋊₁₆D₆ Q₈.₁₅D₃₀ D₄.₁₀D₃₀
Q₈×D₁₅ is a maximal quotient of
Dic₁₅⋊₁₀Q₈ C₄⋊Dic₃₀ Dic₁₅.₃Q₈ D₃₀⋊₅Q₈ D₃₀⋊₆Q₈ Dic₁₅⋊₄Q₈ D₃₀⋊₇Q₈

45 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	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	3	4	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	15	15	2	2	2	2	30	30	30	2	2	2	2	2	4	4	4	2	2	2	2	4	···	4	2	2	2	2	4	···	4

45 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	-	+	+	+	+	+	-	-	-
image	C₁	C₂	C₂	C₂	S₃	Q₈	D₅	D₆	D₁₀	D₁₅	D₃₀	S₃×Q₈	Q₈×D₅	Q₈×D₁₅
kernel	Q₈×D₁₅	Dic₃₀	C₄×D₁₅	Q₈×C₁₅	C₅×Q₈	D₁₅	C₃×Q₈	C₂₀	C₁₂	Q₈	C₄	C₅	C₃	C₁
# reps	1	3	3	1	1	2	2	3	6	4	12	1	2	4

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

1	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	60	59
0	0	0	0	1	1

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	17	38
0	0	0	0	2	44

43	17	0	0	0	0
43	0	0	0	0	0
0	0	0	1	0	0
0	0	60	60	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,59,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,2,0,0,0,0,38,44],[43,43,0,0,0,0,17,0,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

Q₈×D₁₅ in GAP, Magma, Sage, TeX

Q_8\times D_{15}

% in TeX

G:=Group("Q8xD15");

// GroupNames label

G:=SmallGroup(240,181);

// by ID

G=gap.SmallGroup(240,181);

# by ID

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

// Polycyclic

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

// generators/relations

G = Q8×D15 order 240 = 24·3·5

Direct product of Q8 and D15

G = Q₈×D₁₅ order 240 = 2⁴·3·5

Direct product of Q₈ and D₁₅