Dic5.D6 - GroupNames

G = Dic₅.D₆ order 240 = 2⁴·3·5

5^th non-split extension by Dic₅ of D₆ acting via D₆/S₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀.₃D₆, Dic₅.₅D₆, C₃₀.₁₆C₂³, D₃₀.₃C₂², Dic₃.₁₂D₁₀, Dic₁₅.₅C₂², C₁₅:Q₈:₄C₂, C₅:D₄:₃S₃, C₃:₄(C₄oD₂₀), C₁₅:₈(C₄oD₄), C₃:D₂₀:₄C₂, C₁₅:₇D₄:₃C₂, (C₂xC₆).₁D₁₀, C₅:₃(D₄:₂S₃), D₃₀.C₂:₃C₂, (C₂xDic₃):₃D₅, (D₅xDic₃):₃C₂, (C₂xC₁₀).₁₃D₆, C₂².₁(S₃xD₅), (C₁₀xDic₃):₄C₂, (C₆xD₅).₃C₂², C₆.₁₆(C₂²xD₅), C₁₀.₁₆(C₂²xS₃), (C₂xC₃₀).₁₀C₂², (C₃xDic₅).₅C₂², (C₅xDic₃).₁₁C₂², C₂.₁₈(C₂xS₃xD₅), (C₃xC₅:D₄):₁C₂, SmallGroup(240,140)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — Dic₅.D₆

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — D₅xDic₃ — Dic₅.D₆

Lower central

C₁₅ — C₃₀ — Dic₅.D₆

Upper central

C₁ — C₂ — C₂²

Generators and relations for Dic₅.D₆
G = < a,b,c,d | a¹⁰=c⁶=1, b²=d²=a⁵, bab^-1=cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=a⁵b, dcd^-1=c^-1 >

Subgroups: 344 in 80 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₂xC₄, D₄, Q₈, D₅, C₁₀, C₁₀, Dic₃, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₁₅, C₄oD₄, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂xC₁₀, Dic₆, C₄xS₃, C₂xDic₃, C₂xDic₃, C₃:D₄, C₃xD₄, C₃xD₅, D₁₅, C₃₀, C₃₀, Dic₁₀, C₄xD₅, D₂₀, C₅:D₄, C₅:D₄, C₂xC₂₀, D₄:₂S₃, C₅xDic₃, C₃xDic₅, Dic₁₅, C₆xD₅, D₃₀, C₂xC₃₀, C₄oD₂₀, D₅xDic₃, D₃₀.C₂, C₃:D₂₀, C₁₅:Q₈, C₃xC₅:D₄, C₁₀xDic₃, C₁₅:₇D₄, Dic₅.D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄oD₄, D₁₀, C₂²xS₃, C₂²xD₅, D₄:₂S₃, S₃xD₅, C₄oD₂₀, C₂xS₃xD₅, Dic₅.D₆

Smallest permutation representation of Dic₅.D₆
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

Dic₅.D₆ is a maximal subgroup of
D₂₀.₃₈D₆ S₃xC₄oD₂₀ C₁₅:2_-¹⁺⁴ D₅xD₄:₂S₃ D₃₀.C₂³ D₂₀:₁₄D₆ C₁₅:2₊¹⁺⁴
Dic₅.D₆ is a maximal quotient of
Dic₃:₅Dic₁₀ Dic₅.₂Dic₆ Dic₁₅.Q₈ C₄:Dic₃:D₅ Dic₃.D₂₀ D₃₀.D₄ (C₂xC₁₂).D₁₀ (C₄xDic₃):D₅ Dic₃.₃Dic₁₀ C₁₀.D₄:S₃ (D₅xDic₃):C₄ Dic₃:₄D₂₀ D₃₀.₂₃(C₂xC₄) D₁₀.₁₆D₁₂ D₃₀:₂Q₈ Dic₁₅.D₄ Dic₁₅.₁₉D₄ C₂³.₂₆(S₃xD₅) C₂³.₁₃(S₃xD₅) C₂³.₁₄(S₃xD₅) D₃₀:₆D₄ C₆.(D₄xD₅) (C₂xC₃₀).D₄ C₁₀.(C₂xD₁₂) C₂³.₁₇(S₃xD₅) Dic₁₅:₃D₄ Dic₃xC₅:D₄ C₁₅:₂₈(C₄xD₄) D₃₀.₁₆D₄ (C₂xC₆):D₂₀ (C₂xC₁₀):₈Dic₆

39 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	5A	5B	6A	6B	6C	10A	···	10F	12	15A	15B	20A	···	20H	30A	···	30F
order	1	2	2	2	2	3	4	4	4	4	4	5	5	6	6	6	10	···	10	12	15	15	20	···	20	30	···	30
size	1	1	2	10	30	2	3	3	6	10	30	2	2	2	4	20	2	···	2	20	4	4	6	···	6	4	···	4

39 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+		-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₅	D₆	D₆	D₆	C₄oD₄	D₁₀	D₁₀	C₄oD₂₀	D₄:₂S₃	S₃xD₅	C₂xS₃xD₅	Dic₅.D₆
kernel	Dic₅.D₆	D₅xDic₃	D₃₀.C₂	C₃:D₂₀	C₁₅:Q₈	C₃xC₅:D₄	C₁₀xDic₃	C₁₅:₇D₄	C₅:D₄	C₂xDic₃	Dic₅	D₁₀	C₂xC₁₀	C₁₅	Dic₃	C₂xC₆	C₃	C₅	C₂²	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	2	1	1	1	2	4	2	8	1	2	2	4

Matrix representation of Dic₅.D₆ ►in GL₆(F₆₁)

60	0	0	0	0	0
0	60	0	0	0	0
0	0	44	1	0	0
0	0	16	60	0	0
0	0	0	0	1	0
0	0	0	0	0	1

50	0	0	0	0	0
27	11	0	0	0	0
0	0	44	43	0	0
0	0	16	17	0	0
0	0	0	0	60	0
0	0	0	0	0	60

11	18	0	0	0	0
34	50	0	0	0	0
0	0	17	18	0	0
0	0	45	44	0	0
0	0	0	0	60	60
0	0	0	0	1	0

60	15	0	0	0	0
8	1	0	0	0	0
0	0	44	43	0	0
0	0	16	17	0	0
0	0	0	0	1	1
0	0	0	0	0	60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,27,0,0,0,0,0,11,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[11,34,0,0,0,0,18,50,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[60,8,0,0,0,0,15,1,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,1,0,0,0,0,0,1,60] >;

Dic₅.D₆ in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_6

% in TeX

G:=Group("Dic5.D6");

// GroupNames label

G:=SmallGroup(240,140);

// by ID

G=gap.SmallGroup(240,140);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,490,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^10=c^6=1,b^2=d^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;

// generators/relations

G = Dic5.D6 order 240 = 24·3·5

5th non-split extension by Dic5 of D6 acting via D6/S3=C2

G = Dic₅.D₆ order 240 = 2⁴·3·5

5^th non-split extension by Dic₅ of D₆ acting via D₆/S₃=C₂