C2^3xD15 - GroupNames

G = C₂³×D₁₅ order 240 = 2⁴·3·5

Direct product of C₂³ and D₁₅

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

Aliases: C₂³×D₁₅, C₁₅⋊₂C₂⁴, C₃₀⋊₂C₂³, (C₂×C₆)⋊₉D₁₀, C₅⋊₂(S₃×C₂³), C₃⋊₂(C₂³×D₅), (C₂×C₁₀)⋊₁₂D₆, C₆⋊₂(C₂²×D₅), (C₂²×C₆)⋊₃D₅, (C₂²×C₁₀)⋊₅S₃, (C₂²×C₃₀)⋊₃C₂, C₁₀⋊₂(C₂²×S₃), (C₂×C₃₀)⋊₁₀C₂², SmallGroup(240,207)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₂³×D₁₅

Chief series

C₁ — C₅ — C₁₅ — D₁₅ — D₃₀ — C₂²×D₁₅ — C₂³×D₁₅

Lower central

C₁₅ — C₂³×D₁₅

Upper central

C₁ — C₂³

Generators and relations for C₂³×D₁₅
G = < a,b,c,d,e | a²=b²=c²=d¹⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 1288 in 268 conjugacy classes, 115 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₅, S₃, C₆, C₂³, C₂³, D₅, C₁₀, D₆, C₂×C₆, C₁₅, C₂⁴, D₁₀, C₂×C₁₀, C₂²×S₃, C₂²×C₆, D₁₅, C₃₀, C₂²×D₅, C₂²×C₁₀, S₃×C₂³, D₃₀, C₂×C₃₀, C₂³×D₅, C₂²×D₁₅, C₂²×C₃₀, C₂³×D₁₅
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₂⁴, D₁₀, C₂²×S₃, D₁₅, C₂²×D₅, S₃×C₂³, D₃₀, C₂³×D₅, C₂²×D₁₅, C₂³×D₁₅

Smallest permutation representation of C₂³×D₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

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

C₂³×D₁₅ is a maximal subgroup of D₃₀.₄₅D₄ D₃₀⋊₁₈D₄ D₃₀⋊₁₉D₄ D₃₀⋊₁₆D₄ D₃₀⋊₁₇D₄ S₃×C₂³×D₅
C₂³×D₁₅ is a maximal quotient of D₄⋊₆D₃₀ Q₈.₁₅D₃₀ D₄⋊₈D₃₀ D₄.₁₀D₃₀

72 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3	5A	5B	6A	···	6G	10A	···	10N	15A	15B	15C	15D	30A	···	30AB
order	1	2	···	2	2	···	2	3	5	5	6	···	6	10	···	10	15	15	15	15	30	···	30
size	1	1	···	1	15	···	15	2	2	2	2	···	2	2	···	2	2	2	2	2	2	···	2

72 irreducible representations

dim	1	1	1	2	2	2	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	D₅	D₆	D₁₀	D₁₅	D₃₀
kernel	C₂³×D₁₅	C₂²×D₁₅	C₂²×C₃₀	C₂²×C₁₀	C₂²×C₆	C₂×C₁₀	C₂×C₆	C₂³	C₂²
# reps	1	14	1	1	2	7	14	4	28

Matrix representation of C₂³×D₁₅ ►in GL₅(𝔽₃₁)

30	0	0	0	0
0	30	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	30

30	0	0	0	0
0	30	0	0	0
0	0	30	0	0
0	0	0	1	0
0	0	0	0	1

30	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	30	0
0	0	0	0	30

1	0	0	0	0
0	30	1	0	0
0	30	0	0	0
0	0	0	18	12
0	0	0	18	0

30	0	0	0	0
0	30	0	0	0
0	30	1	0	0
0	0	0	30	1
0	0	0	0	1

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1],[30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,30,30,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,12,0],[30,0,0,0,0,0,30,30,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,1,1] >;

C₂³×D₁₅ in GAP, Magma, Sage, TeX

C_2^3\times D_{15}

% in TeX

G:=Group("C2^3xD15");

// GroupNames label

G:=SmallGroup(240,207);

// by ID

G=gap.SmallGroup(240,207);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23×D15 order 240 = 24·3·5

Direct product of C23 and D15

G = C₂³×D₁₅ order 240 = 2⁴·3·5

Direct product of C₂³ and D₁₅