C2xC5:D12 - GroupNames

G = C₂xC₅:D₁₂ order 240 = 2⁴·3·5

Direct product of C₂ and C₅:D₁₂

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

Aliases: C₂xC₅:D₁₂, C₃₀:₃D₄, D₆:₆D₁₀, C₁₀:₂D₁₂, Dic₅:₄D₆, D₃₀:₉C₂², C₃₀.₂₃C₂³, C₁₅:₆(C₂xD₄), C₅:₃(C₂xD₁₂), C₆:₁(C₅:D₄), (C₂xDic₅):₄S₃, (C₆xDic₅):₆C₂, (C₂xC₁₀).₁₈D₆, (C₂xC₆).₁₈D₁₀, (C₂²xS₃):₂D₅, (S₃xC₁₀):₆C₂², (C₂²xD₁₅):₅C₂, C₂².₁₆(S₃xD₅), C₆.₂₃(C₂²xD₅), C₁₀.₂₃(C₂²xS₃), (C₂xC₃₀).₁₇C₂², (C₃xDic₅):₇C₂², C₃:₁(C₂xC₅:D₄), (S₃xC₂xC₁₀):₂C₂, C₂.₂₃(C₂xS₃xD₅), SmallGroup(240,147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₂xC₅:D₁₂

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xDic₅ — C₅:D₁₂ — C₂xC₅:D₁₂

Lower central

C₁₅ — C₃₀ — C₂xC₅:D₁₂

Upper central

C₁ — C₂²

Generators and relations for C₂xC₅:D₁₂
G = < a,b,c,d | a²=b⁵=c¹²=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 528 in 108 conjugacy classes, 40 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₂xC₄, D₄, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₁₂, D₆, D₆, C₂xC₆, C₁₅, C₂xD₄, Dic₅, D₁₀, C₂xC₁₀, C₂xC₁₀, D₁₂, C₂xC₁₂, C₂²xS₃, C₂²xS₃, C₅xS₃, D₁₅, C₃₀, C₃₀, C₂xDic₅, C₅:D₄, C₂²xD₅, C₂²xC₁₀, C₂xD₁₂, C₃xDic₅, S₃xC₁₀, S₃xC₁₀, D₃₀, D₃₀, C₂xC₃₀, C₂xC₅:D₄, C₅:D₁₂, C₆xDic₅, S₃xC₂xC₁₀, C₂²xD₁₅, C₂xC₅:D₁₂
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂xD₄, D₁₀, D₁₂, C₂²xS₃, C₅:D₄, C₂²xD₅, C₂xD₁₂, S₃xD₅, C₂xC₅:D₄, C₅:D₁₂, C₂xS₃xD₅, C₂xC₅:D₁₂

Smallest permutation representation of C₂xC₅:D₁₂
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

C₂xC₅:D₁₂ is a maximal subgroup of
Dic₅.D₁₂ (C₂xC₂₀).D₆ Dic₅.₈D₁₂ D₆:Dic₅:C₂ D₃₀.₃₅D₄ Dic₅:₄D₁₂ Dic₁₅:₁₄D₄ Dic₅:D₁₂ D₃₀:D₄ D₆.D₂₀ D₃₀.₇D₄ C₁₅:₂₂(C₄xD₄) D₁₀:D₁₂ C₂₀:D₁₂ D₆:D₂₀ C₆₀:₆D₄ D₃₀:₁₂D₄ C₂₀:₂D₁₂ D₃₀:₄D₄ D₃₀:₅D₄ D₃₀:₇D₄ Dic₁₅:₄D₄ (C₂xC₁₀):₄D₁₂ Dic₁₅:₅D₄ (C₂xC₁₀):₁₁D₁₂ D₃₀:₁₉D₄ C₂xD₅xD₁₂ D₁₂:₁₄D₁₀ C₂xS₃xC₅:D₄
C₂xC₅:D₁₂ is a maximal quotient of
C₂₀.₆₀D₁₂ D₆₀:₃₆C₂² C₆₀.₃₈D₄ C₂₀.D₁₂ D₁₂.₃₃D₁₀ C₆₀.₄₅D₄ C₆₀.₆₉D₄ C₆₀.₇₀D₄ Dic₅:Dic₆ D₃₀:₉Q₈ C₂₀:D₁₂ C₆₀:₆D₄ C₂₀:₂D₁₂ C₁₀.(C₂xD₁₂) (C₂xC₁₀).D₁₂ (C₂xC₁₀):₄D₁₂ (C₂xC₁₀):₁₁D₁₂ D₃₀:₁₉D₄

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	5A	5B	6A	6B	6C	10A	···	10F	10G	···	10N	12A	12B	12C	12D	15A	15B	30A	···	30F
order	1	2	2	2	2	2	2	2	3	4	4	5	5	6	6	6	10	···	10	10	···	10	12	12	12	12	15	15	30	···	30
size	1	1	1	1	6	6	30	30	2	10	10	2	2	2	2	2	2	···	2	6	···	6	10	10	10	10	4	4	4	···	4

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₅

D₆

D₁₀

D₁₂

C₅:D₄

S₃xD₅

C₅:D₁₂

C₂xS₃xD₅

kernel

C₂xC₅:D₁₂

C₅:D₁₂

C₆xDic₅

S₃xC₂xC₁₀

C₂²xD₁₅

C₂xDic₅

C₃₀

C₂²xS₃

Dic₅

C₂xC₁₀

D₆

C₂xC₆

C₁₀

C₆

C₂²

C₂

# reps

Matrix representation of C₂xC₅:D₁₂ ►in GL₄(F₆₁) generated by

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

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

0	60	0	0
1	1	0	0
0	0	31	17
0	0	8	30

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

G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,43,1,0,0,60,0],[0,1,0,0,60,1,0,0,0,0,31,8,0,0,17,30],[0,1,0,0,1,0,0,0,0,0,1,43,0,0,0,60] >;

C₂xC₅:D₁₂ in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{12}

% in TeX

G:=Group("C2xC5:D12");

// GroupNames label

G:=SmallGroup(240,147);

// by ID

G=gap.SmallGroup(240,147);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,121,490,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C2xC5:D12 order 240 = 24·3·5

Direct product of C2 and C5:D12

G = C₂xC₅:D₁₂ order 240 = 2⁴·3·5

Direct product of C₂ and C₅:D₁₂