C2xS3xDic5 - GroupNames

G = C₂xS₃xDic₅ order 240 = 2⁴·3·5

Direct product of C₂, S₃ and Dic₅

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

Aliases: C₂xS₃xDic₅, D₆.₁₁D₁₀, C₃₀.₁₈C₂³, Dic₁₅:₈C₂², C₁₀:₄(C₄xS₃), C₃₀:₅(C₂xC₄), (S₃xC₁₀):₄C₄, C₁₅:₆(C₂²xC₄), C₆:₁(C₂xDic₅), (C₆xDic₅):₃C₂, (C₂xC₁₀).₁₄D₆, (C₂xC₆).₁₃D₁₀, C₃:₁(C₂²xDic₅), (C₂xDic₁₅):₉C₂, (C₂²xS₃).₂D₅, C₂².₁₂(S₃xD₅), C₆.₁₈(C₂²xD₅), C₁₀.₁₈(C₂²xS₃), (C₂xC₃₀).₁₂C₂², (C₃xDic₅):₅C₂², (S₃xC₁₀).₁₁C₂², C₅:₅(S₃xC₂xC₄), C₂.₃(C₂xS₃xD₅), (C₅xS₃):₃(C₂xC₄), (S₃xC₂xC₁₀).₂C₂, SmallGroup(240,142)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₂xS₃xDic₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xDic₅ — S₃xDic₅ — C₂xS₃xDic₅

Lower central

C₁₅ — C₂xS₃xDic₅

Upper central

C₁ — C₂²

Generators and relations for C₂xS₃xDic₅
G = < a,b,c,d,e | a²=b³=c²=d¹⁰=1, e²=d⁵, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 336 in 108 conjugacy classes, 56 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₂xC₄, C₂³, C₁₀, C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₂xC₆, C₁₅, C₂²xC₄, Dic₅, Dic₅, C₂xC₁₀, C₂xC₁₀, C₄xS₃, C₂xDic₃, C₂xC₁₂, C₂²xS₃, C₅xS₃, C₃₀, C₃₀, C₂xDic₅, C₂xDic₅, C₂²xC₁₀, S₃xC₂xC₄, C₃xDic₅, Dic₁₅, S₃xC₁₀, C₂xC₃₀, C₂²xDic₅, S₃xDic₅, C₆xDic₅, C₂xDic₁₅, S₃xC₂xC₁₀, C₂xS₃xDic₅
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, D₅, D₆, C₂²xC₄, Dic₅, D₁₀, C₄xS₃, C₂²xS₃, C₂xDic₅, C₂²xD₅, S₃xC₂xC₄, S₃xD₅, C₂²xDic₅, S₃xDic₅, C₂xS₃xD₅, C₂xS₃xDic₅

Smallest permutation representation of C₂xS₃xDic₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

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

C₂xS₃xDic₅ is a maximal subgroup of
Dic₅.₂₂D₁₂ D₆.(C₄xD₅) (S₃xDic₅):C₄ Dic₅:₄D₁₂ Dic₁₅:₁₄D₄ D₆:₁Dic₁₀ Dic₅:D₁₂ D₆:₂Dic₁₀ (C₂xD₁₂).D₅ D₆.D₂₀ D₆:₃Dic₁₀ Dic₁₅:₈D₄ D₆:₄Dic₁₀ D₆:(C₄xD₅) Dic₁₅:₉D₄ Dic₁₅:₂D₄ D₆.₉D₂₀ (S₃xC₁₀).D₄ Dic₁₅:₄D₄ Dic₁₅:₁₇D₄ (S₃xC₁₀):D₄ S₃xC₂xC₄xD₅
C₂xS₃xDic₅ is a maximal quotient of
D₁₂.₂Dic₅ D₁₂.Dic₅ (S₃xC₂₀):₅C₄ Dic₁₅:₇Q₈ (S₃xC₂₀):₇C₄ Dic₁₅:₈D₄ C₂³.₂₆(S₃xD₅) Dic₁₅:₁₇D₄

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	10A	···	10F	10G	···	10N	12A	12B	12C	12D	15A	15B	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	10	···	10	10	···	10	12	12	12	12	15	15	30	···	30
size	1	1	1	1	3	3	3	3	2	5	5	5	5	15	15	15	15	2	2	2	2	2	2	···	2	6	···	6	10	10	10	10	4	4	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	D₆	D₆	Dic₅	D₁₀	D₁₀	C₄xS₃	S₃xD₅	S₃xDic₅	C₂xS₃xD₅
kernel	C₂xS₃xDic₅	S₃xDic₅	C₆xDic₅	C₂xDic₁₅	S₃xC₂xC₁₀	S₃xC₁₀	C₂xDic₅	C₂²xS₃	Dic₅	C₂xC₁₀	D₆	D₆	C₂xC₆	C₁₀	C₂²	C₂	C₂
# reps	1	4	1	1	1	8	1	2	2	1	8	4	2	4	2	4	2

Matrix representation of C₂xS₃xDic₅ ►in GL₄(F₆₁) generated by

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

1	15	0	0
12	59	0	0
0	0	1	0
0	0	0	1

1	15	0	0
0	60	0	0
0	0	60	0
0	0	0	60

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

50	0	0	0
0	50	0	0
0	0	16	9
0	0	53	45

G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,12,0,0,15,59,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,15,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,43,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,16,53,0,0,9,45] >;

C₂xS₃xDic₅ in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_5

% in TeX

G:=Group("C2xS3xDic5");

// GroupNames label

G:=SmallGroup(240,142);

// by ID

G=gap.SmallGroup(240,142);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^10=1,e^2=d^5,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 = C2xS3xDic5 order 240 = 24·3·5

Direct product of C2, S3 and Dic5

G = C₂xS₃xDic₅ order 240 = 2⁴·3·5

Direct product of C₂, S₃ and Dic₅