C2xC15:D4 - GroupNames

G = C₂xC₁₅:D₄ order 240 = 2⁴·3·5

Direct product of C₂ and C₁₅:D₄

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

Aliases: C₂xC₁₅:D₄, C₃₀:₁D₄, D₆:₅D₁₀, D₁₀:₅D₆, C₃₀.₂₁C₂³, Dic₁₅:₉C₂², C₁₅:₄(C₂xD₄), C₆:₂(C₅:D₄), C₁₀:₂(C₃:D₄), (C₂xC₁₀).₁₆D₆, (C₂xC₆).₁₆D₁₀, (C₂²xD₅):₂S₃, (C₂²xS₃):₁D₅, (C₆xD₅):₅C₂², (S₃xC₁₀):₅C₂², C₂².₁₄(S₃xD₅), C₆.₂₁(C₂²xD₅), (C₂xDic₁₅):₁₀C₂, C₁₀.₂₁(C₂²xS₃), (C₂xC₃₀).₁₅C₂², (D₅xC₂xC₆):₁C₂, C₅:₃(C₂xC₃:D₄), C₃:₃(C₂xC₅:D₄), (S₃xC₂xC₁₀):₁C₂, C₂.₂₁(C₂xS₃xD₅), SmallGroup(240,145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₂xC₁₅:D₄

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — C₁₅:D₄ — C₂xC₁₅:D₄

Lower central

C₁₅ — C₃₀ — C₂xC₁₅:D₄

Upper central

C₁ — C₂²

Generators and relations for C₂xC₁₅:D₄
G = < a,b,c,d | a²=b¹⁵=c⁴=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, dbd=b⁴, dcd=c^-1 >

Subgroups: 432 in 108 conjugacy classes, 40 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₆, C₂xC₄, D₄, C₂³, D₅, C₁₀, C₁₀, C₁₀, Dic₃, D₆, D₆, C₂xC₆, C₂xC₆, C₁₅, C₂xD₄, Dic₅, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xDic₃, C₃:D₄, C₂²xS₃, C₂²xC₆, C₅xS₃, C₃xD₅, C₃₀, C₃₀, C₂xDic₅, C₅:D₄, C₂²xD₅, C₂²xC₁₀, C₂xC₃:D₄, Dic₁₅, C₆xD₅, C₆xD₅, S₃xC₁₀, S₃xC₁₀, C₂xC₃₀, C₂xC₅:D₄, C₁₅:D₄, C₂xDic₁₅, D₅xC₂xC₆, S₃xC₂xC₁₀, C₂xC₁₅:D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂xD₄, D₁₀, C₃:D₄, C₂²xS₃, C₅:D₄, C₂²xD₅, C₂xC₃:D₄, S₃xD₅, C₂xC₅:D₄, C₁₅:D₄, C₂xS₃xD₅, C₂xC₁₅:D₄

Smallest permutation representation of C₂xC₁₅:D₄
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

C₂xC₁₅:D₄ is a maximal subgroup of
D₁₀.D₁₂ Dic₁₅:D₄ D₁₀.₁₇D₁₂ D₆:(C₄xD₅) C₁₅:₁₇(C₄xD₄) Dic₁₅:₉D₄ D₆:C₄:D₅ D₁₀:D₁₂ C₆₀:D₄ D₁₀:C₄:S₃ Dic₁₅:₂D₄ D₆:D₂₀ C₆₀:₄D₄ D₆.₉D₂₀ D₃₀:₁₂D₄ Dic₁₅.₁₀D₄ C₆₀:₁₀D₄ Dic₁₅.₃₁D₄ D₆:₄D₂₀ D₃₀:₄D₄ (C₆xD₅):D₄ (C₂xC₃₀):D₄ (S₃xC₁₀):D₄ Dic₁₅:₅D₄ C₁₅:C₂²wrC₂ Dic₁₅:₁₈D₄ D₂₀:₁₃D₆ C₂xD₅xC₃:D₄ C₂xS₃xC₅:D₄
C₂xC₁₅:D₄ is a maximal quotient of
D₂₀.₃₄D₆ C₆₀.₃₆D₄ D₂₀:₂₁D₆ D₂₀.₃₇D₆ D₁₂.₃₇D₁₀ C₆₀.₆₇D₄ C₆₀.₈₈D₄ C₆₀.₄₆D₄ C₆₀.₈₉D₄ Dic₁₅:₈Q₈ C₆₀:D₄ C₆₀:₄D₄ C₆₀:₁₀D₄ (C₂xC₃₀).D₄ C₃₀.(C₂xD₄) (C₂xC₃₀):D₄ C₁₅:C₂²wrC₂ Dic₁₅:₁₈D₄

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	···	10F	10G	···	10N	15A	15B	30A	···	30F
order	1	2	2	2	2	2	2	2	3	4	4	5	5	6	6	6	6	6	6	6	10	···	10	10	···	10	15	15	30	···	30
size	1	1	1	1	6	6	10	10	2	30	30	2	2	2	2	2	10	10	10	10	2	···	2	6	···	6	4	4	4	···	4

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₅

D₆

D₁₀

C₃:D₄

C₅:D₄

S₃xD₅

C₁₅:D₄

C₂xS₃xD₅

kernel

C₂xC₁₅:D₄

C₁₅:D₄

C₂xDic₁₅

D₅xC₂xC₆

S₃xC₂xC₁₀

C₂²xD₅

C₃₀

C₂²xS₃

D₁₀

C₂xC₁₀

D₆

C₂xC₆

C₁₀

C₆

C₂²

C₂

# reps

Matrix representation of C₂xC₁₅:D₄ ►in GL₅(F₆₁)

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

1	0	0	0	0
0	47	0	0	0
0	27	13	0	0
0	0	0	44	44
0	0	0	17	60

1	0	0	0	0
0	15	15	0	0
0	50	46	0	0
0	0	0	1	0
0	0	0	17	60

1	0	0	0	0
0	60	0	0	0
0	2	1	0	0
0	0	0	1	0
0	0	0	17	60

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,47,27,0,0,0,0,13,0,0,0,0,0,44,17,0,0,0,44,60],[1,0,0,0,0,0,15,50,0,0,0,15,46,0,0,0,0,0,1,17,0,0,0,0,60],[1,0,0,0,0,0,60,2,0,0,0,0,1,0,0,0,0,0,1,17,0,0,0,0,60] >;

C₂xC₁₅:D₄ in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes D_4

% in TeX

G:=Group("C2xC15:D4");

// GroupNames label

G:=SmallGroup(240,145);

// by ID

G=gap.SmallGroup(240,145);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,490,6917]);

// Polycyclic

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

// generators/relations

G = C2xC15:D4 order 240 = 24·3·5

Direct product of C2 and C15:D4

G = C₂xC₁₅:D₄ order 240 = 2⁴·3·5

Direct product of C₂ and C₁₅:D₄