C2xD5xDic3 - GroupNames

G = C₂xD₅xDic₃ order 240 = 2⁴·3·5

Direct product of C₂, D₅ and Dic₃

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

Aliases: C₂xD₅xDic₃, D₁₀.₁₉D₆, C₃₀.₁₅C₂³, Dic₁₅:₇C₂², C₆:₃(C₄xD₅), C₃₀:₄(C₂xC₄), (C₆xD₅):₂C₄, C₁₅:₅(C₂²xC₄), C₁₀:₂(C₂xDic₃), (C₂xC₆).₁₂D₁₀, (C₂xC₁₀).₁₂D₆, C₅:₂(C₂²xDic₃), (C₁₀xDic₃):₃C₂, (C₂xDic₁₅):₇C₂, (C₂xC₃₀).₉C₂², (C₂²xD₅).₅S₃, C₂².₁₁(S₃xD₅), C₆.₁₅(C₂²xD₅), C₁₀.₁₅(C₂²xS₃), (C₅xDic₃):₅C₂², (C₆xD₅).₁₅C₂², C₃:₄(C₂xC₄xD₅), C₂.₂(C₂xS₃xD₅), (D₅xC₂xC₆).₂C₂, (C₃xD₅):₃(C₂xC₄), SmallGroup(240,139)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₂xD₅xDic₃

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — D₅xDic₃ — C₂xD₅xDic₃

Lower central

C₁₅ — C₂xD₅xDic₃

Upper central

C₁ — C₂²

Generators and relations for C₂xD₅xDic₃
G = < a,b,c,d,e | a²=b⁵=c²=d⁶=1, e²=d³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 368 in 108 conjugacy classes, 56 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₆, C₂xC₄, C₂³, D₅, C₁₀, C₁₀, Dic₃, Dic₃, C₂xC₆, C₂xC₆, C₁₅, C₂²xC₄, Dic₅, C₂₀, D₁₀, C₂xC₁₀, C₂xDic₃, C₂xDic₃, C₂²xC₆, C₃xD₅, C₃₀, C₃₀, C₄xD₅, C₂xDic₅, C₂xC₂₀, C₂²xD₅, C₂²xDic₃, C₅xDic₃, Dic₁₅, C₆xD₅, C₂xC₃₀, C₂xC₄xD₅, D₅xDic₃, C₁₀xDic₃, C₂xDic₁₅, D₅xC₂xC₆, C₂xD₅xDic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₅, Dic₃, D₆, C₂²xC₄, D₁₀, C₂xDic₃, C₂²xS₃, C₄xD₅, C₂²xD₅, C₂²xDic₃, S₃xD₅, C₂xC₄xD₅, D₅xDic₃, C₂xS₃xD₅, C₂xD₅xDic₃

Smallest permutation representation of C₂xD₅xDic₃
►On 120 points

Generators in S₁₂₀

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

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

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

(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)

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

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

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

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

C₂xD₅xDic₃ is a maximal subgroup of
D₁₀.₂₀D₁₂ (D₅xDic₃):C₄ D₁₀.₁₉(C₄xS₃) Dic₃:₄D₂₀ Dic₁₅:₁₃D₄ (C₆xD₅).D₄ Dic₁₅:D₄ Dic₃:D₂₀ D₁₀.₁₆D₁₂ D₁₀.₁₇D₁₂ D₁₀:₁Dic₆ D₁₀:₂Dic₆ Dic₁₅.D₄ D₁₀:₄Dic₆ D₂₀:₈Dic₃ C₁₅:₁₇(C₄xD₄) Dic₁₅:₉D₄ C₂³.₁₇(S₃xD₅) (C₆xD₅):D₄ Dic₁₅:₃D₄ Dic₁₅:₁₆D₄ C₂²:F₅.S₃ S₃xC₂xC₄xD₅
C₂xD₅xDic₃ is a maximal quotient of
D₂₀.₃Dic₃ D₂₀.₂Dic₃ Dic₁₅:₆Q₈ (D₅xC₁₂):C₄ (C₄xD₅):Dic₃ D₂₀:₈Dic₃ (C₆xDic₅):₇C₄ Dic₁₅:₁₆D₄

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	···	10F	15A	15B	20A	···	20H	30A	···	30F
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	5	5	6	6	6	6	6	6	6	10	···	10	15	15	20	···	20	30	···	30
size	1	1	1	1	5	5	5	5	2	3	3	3	3	15	15	15	15	2	2	2	2	2	10	10	10	10	2	···	2	4	4	6	···	6	4	···	4

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+		+	+	-	+	+	+	+		+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₄	S₃	D₅	Dic₃	D₆	D₆	D₁₀	D₁₀	C₄xD₅	S₃xD₅	D₅xDic₃	C₂xS₃xD₅
kernel	C₂xD₅xDic₃	D₅xDic₃	C₁₀xDic₃	C₂xDic₁₅	D₅xC₂xC₆	C₆xD₅	C₂²xD₅	C₂xDic₃	D₁₀	D₁₀	C₂xC₁₀	Dic₃	C₂xC₆	C₆	C₂²	C₂	C₂
# reps	1	4	1	1	1	8	1	2	4	2	1	4	2	8	2	4	2

Matrix representation of C₂xD₅xDic₃ ►in GL₅(F₆₁)

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

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	60	1
0	0	0	42	18

60	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	19	60

60	0	0	0	0
0	59	46	0	0
0	49	1	0	0
0	0	0	60	0
0	0	0	0	60

11	0	0	0	0
0	4	48	0	0
0	34	57	0	0
0	0	0	11	0
0	0	0	0	11

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,42,0,0,0,1,18],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,19,0,0,0,0,60],[60,0,0,0,0,0,59,49,0,0,0,46,1,0,0,0,0,0,60,0,0,0,0,0,60],[11,0,0,0,0,0,4,34,0,0,0,48,57,0,0,0,0,0,11,0,0,0,0,0,11] >;

C₂xD₅xDic₃ in GAP, Magma, Sage, TeX

C_2\times D_5\times {\rm Dic}_3

% in TeX

G:=Group("C2xD5xDic3");

// GroupNames label

G:=SmallGroup(240,139);

// by ID

G=gap.SmallGroup(240,139);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,490,6917]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2xD5xDic3 order 240 = 24·3·5

Direct product of C2, D5 and Dic3

G = C₂xD₅xDic₃ order 240 = 2⁴·3·5

Direct product of C₂, D₅ and Dic₃