C2xC6xD15 - GroupNames

G = C₂xC₆xD₁₅ order 360 = 2³·3²·5

Direct product of C₂xC₆ and D₁₅

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

Aliases: C₂xC₆xD₁₅, C₃₀:₇D₆, C₆²:₄D₅, C₆:₂(C₆xD₅), C₁₀:₂(S₃xC₆), (C₆xC₃₀):₅C₂, C₃₀:₂(C₂xC₆), (C₂xC₃₀):₇C₆, (C₂xC₃₀):₉S₃, (C₃xC₆):₅D₁₀, (C₃xC₁₅):₈C₂³, C₁₅:₈(C₂²xS₃), C₁₅:₂(C₂²xC₆), (C₃xC₃₀):₇C₂², C₃²:₆(C₂²xD₅), C₅:₂(S₃xC₂xC₆), C₃:₂(D₅xC₂xC₆), (C₂xC₆):₅(C₃xD₅), (C₂xC₁₀):₇(C₃xS₃), SmallGroup(360,159)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₂xC₆xD₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃xC₁₅ — C₃xD₁₅ — C₆xD₁₅ — C₂xC₆xD₁₅

Lower central

C₁₅ — C₂xC₆xD₁₅

Upper central

C₁ — C₂xC₆

Generators and relations for C₂xC₆xD₁₅
G = < a,b,c,d | a²=b⁶=c¹⁵=d²=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 588 in 138 conjugacy classes, 62 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², C₅, S₃, C₆, C₆, C₂³, C₃², D₅, C₁₀, D₆, C₂xC₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, D₁₀, C₂xC₁₀, C₂²xS₃, C₂²xC₆, C₃xD₅, D₁₅, C₃₀, C₃₀, S₃xC₆, C₆², C₂²xD₅, C₃xC₁₅, C₆xD₅, D₃₀, C₂xC₃₀, C₂xC₃₀, S₃xC₂xC₆, C₃xD₁₅, C₃xC₃₀, D₅xC₂xC₆, C₂²xD₁₅, C₆xD₁₅, C₆xC₃₀, C₂xC₆xD₁₅
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₅, D₆, C₂xC₆, C₃xS₃, D₁₀, C₂²xS₃, C₂²xC₆, C₃xD₅, D₁₅, S₃xC₆, C₂²xD₅, C₆xD₅, D₃₀, S₃xC₂xC₆, C₃xD₁₅, D₅xC₂xC₆, C₂²xD₁₅, C₆xD₁₅, C₂xC₆xD₁₅

Smallest permutation representation of C₂xC₆xD₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

108 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	5A	5B	6A	···	6F	6G	···	6O	6P	···	6W	10A	···	10F	15A	···	15P	30A	···	30AV
order	1	2	2	2	2	2	2	2	3	3	3	3	3	5	5	6	···	6	6	···	6	6	···	6	10	···	10	15	···	15	30	···	30
size	1	1	1	1	15	15	15	15	1	1	2	2	2	2	2	1	···	1	2	···	2	15	···	15	2	···	2	2	···	2	2	···	2

108 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+				+	+	+		+		+			+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₅	D₆	C₃xS₃	D₁₀	C₃xD₅	D₁₅	S₃xC₆	C₆xD₅	D₃₀	C₃xD₁₅	C₆xD₁₅
kernel	C₂xC₆xD₁₅	C₆xD₁₅	C₆xC₃₀	C₂²xD₁₅	D₃₀	C₂xC₃₀	C₂xC₃₀	C₆²	C₃₀	C₂xC₁₀	C₃xC₆	C₂xC₆	C₂xC₆	C₁₀	C₆	C₆	C₂²	C₂
# reps	1	6	1	2	12	2	1	2	3	2	6	4	4	6	12	12	8	24

Matrix representation of C₂xC₆xD₁₅ ►in GL₄(F₃₁) generated by

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

30	0	0	0
0	30	0	0
0	0	5	0
0	0	0	5

30	18	0	0
13	13	0	0
0	0	20	20
0	0	0	14

27	22	0	0
12	4	0	0
0	0	17	20
0	0	29	14

G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[30,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[30,13,0,0,18,13,0,0,0,0,20,0,0,0,20,14],[27,12,0,0,22,4,0,0,0,0,17,29,0,0,20,14] >;

C₂xC₆xD₁₅ in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{15}

% in TeX

G:=Group("C2xC6xD15");

// GroupNames label

G:=SmallGroup(360,159);

// by ID

G=gap.SmallGroup(360,159);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,1444,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = C2xC6xD15 order 360 = 23·32·5

Direct product of C2xC6 and D15

G = C₂xC₆xD₁₅ order 360 = 2³·3²·5

Direct product of C₂xC₆ and D₁₅