Dic10:Dic3 - GroupNames

G = Dic₁₀⋊Dic₃ order 480 = 2⁵·3·5

1^st semidirect product of Dic₁₀ and Dic₃ acting via Dic₃/C₃=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₁₀⋊₁Dic₃, C₁₅⋊₁₁C₄≀C₂, D₄⋊₂(C₃⋊F₅), (C₃×D₄)⋊₂F₅, (D₄×C₁₅)⋊₂C₄, C₆₀.₃₄(C₂×C₄), C₃⋊₃(D₄⋊F₅), (C₅×D₄)⋊₂Dic₃, (C₆×D₅).₃₃D₄, (C₄×D₅).₂₈D₆, C₁₂.₁₀(C₂×F₅), C₁₂.F₅⋊₇C₂, (C₃×Dic₁₀)⋊₁C₄, D₄⋊₂D₅.₂S₃, C₂₀.₂(C₂×Dic₃), C₅⋊₁(Q₈⋊₃Dic₃), D₁₀.₆(C₃⋊D₄), (C₃×Dic₅).₈₂D₄, C₆.₂₁(C₂²⋊F₅), C₃₀.₂₁(C₂²⋊C₄), (D₅×C₁₂).₆₉C₂², Dic₅.₃₇(C₃⋊D₄), C₁₀.₆(C₆.D₄), C₂.₇(D₁₀.D₆), (C₄×C₃⋊F₅)⋊₆C₂, C₄.₂(C₂×C₃⋊F₅), (C₃×D₄⋊₂D₅).₂C₂, SmallGroup(480,313)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — Dic₁₀⋊Dic₃

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃×Dic₅ — D₅×C₁₂ — C₁₂.F₅ — Dic₁₀⋊Dic₃

Lower central

C₁₅ — C₃₀ — C₆₀ — Dic₁₀⋊Dic₃

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for Dic₁₀⋊Dic₃
G = < a,b,c,d | a²⁰=c⁶=1, b²=a¹⁰, d²=c³, bab^-1=a^-1, cac^-1=a⁹, dad^-1=a¹³, cbc^-1=a¹⁰b, dbd^-1=a¹⁵b, dcd^-1=c^-1 >

Subgroups: 460 in 88 conjugacy classes, 29 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, C₆, C₆, C₈, C₂×C₄, D₄, D₄, Q₈, D₅, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, C₂×C₆, C₁₅, C₄², M₄(2), C₄○D₄, Dic₅, Dic₅, C₂₀, F₅, D₁₀, C₂×C₁₀, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×D₅, C₃₀, C₃₀, C₄≀C₂, C₅⋊C₈, Dic₁₀, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₅×D₄, C₂×F₅, C₄.Dic₃, C₄×Dic₃, C₃×C₄○D₄, C₃×Dic₅, C₃×Dic₅, C₆₀, C₃⋊F₅, C₆×D₅, C₂×C₃₀, C₄.F₅, C₄×F₅, D₄⋊₂D₅, Q₈⋊₃Dic₃, C₁₅⋊C₈, C₃×Dic₁₀, D₅×C₁₂, C₆×Dic₅, C₃×C₅⋊D₄, D₄×C₁₅, C₂×C₃⋊F₅, D₄⋊F₅, C₁₂.F₅, C₄×C₃⋊F₅, C₃×D₄⋊₂D₅, Dic₁₀⋊Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, Dic₃, D₆, C₂²⋊C₄, F₅, C₂×Dic₃, C₃⋊D₄, C₄≀C₂, C₂×F₅, C₆.D₄, C₃⋊F₅, C₂²⋊F₅, Q₈⋊₃Dic₃, C₂×C₃⋊F₅, D₄⋊F₅, D₁₀.D₆, Dic₁₀⋊Dic₃

Smallest permutation representation of Dic₁₀⋊Dic₃
►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 70 11 80)(2 69 12 79)(3 68 13 78)(4 67 14 77)(5 66 15 76)(6 65 16 75)(7 64 17 74)(8 63 18 73)(9 62 19 72)(10 61 20 71)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(81 109 91 119)(82 108 92 118)(83 107 93 117)(84 106 94 116)(85 105 95 115)(86 104 96 114)(87 103 97 113)(88 102 98 112)(89 101 99 111)(90 120 100 110)

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	5	6A	6B	6C	6D	8A	8B	10A	10B	10C	12A	12B	12C	12D	12E	15A	15B	20	30A	30B	30C	30D	30E	30F	60A	60B
order	1	2	2	2	3	4	4	4	4	4	4	4	4	5	6	6	6	6	8	8	10	10	10	12	12	12	12	12	15	15	20	30	30	30	30	30	30	60	60
size	1	1	4	10	2	2	5	5	20	30	30	30	30	4	2	4	4	20	60	60	4	8	8	4	10	10	20	20	4	4	8	4	4	8	8	8	8	8	8

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	8	8
type	+	+	+	+			+	+	+	-	+	-				+	+		+				-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	Dic₃	D₆	Dic₃	C₃⋊D₄	C₃⋊D₄	C₄≀C₂	F₅	C₂×F₅	C₃⋊F₅	C₂²⋊F₅	Q₈⋊₃Dic₃	C₂×C₃⋊F₅	D₁₀.D₆	D₄⋊F₅	Dic₁₀⋊Dic₃
kernel	Dic₁₀⋊Dic₃	C₁₂.F₅	C₄×C₃⋊F₅	C₃×D₄⋊₂D₅	C₃×Dic₁₀	D₄×C₁₅	D₄⋊₂D₅	C₃×Dic₅	C₆×D₅	Dic₁₀	C₄×D₅	C₅×D₄	Dic₅	D₁₀	C₁₅	C₃×D₄	C₁₂	D₄	C₆	C₅	C₄	C₂	C₃	C₁
# reps	1	1	1	1	2	2	1	1	1	1	1	1	2	2	4	1	1	2	2	2	2	4	1	2

Matrix representation of Dic₁₀⋊Dic₃ ►in GL₆(𝔽₂₄₁)

177	0	0	0	0	0
0	64	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	240	240	240	240

0	64	0	0	0	0
64	0	0	0	0	0
0	0	1	0	0	0
0	0	240	240	240	240
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	240	0	0	0	0
0	0	126	0	12	12
0	0	115	127	127	115
0	0	12	12	0	126
0	0	0	229	114	229

240	0	0	0	0	0
0	64	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	240	240	240	240

G:=sub<GL(6,GF(241))| [177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,64,0,0,0,0,64,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,126,115,12,0,0,0,0,127,12,229,0,0,12,127,0,114,0,0,12,115,126,229],[240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

Dic₁₀⋊Dic₃ in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes {\rm Dic}_3

% in TeX

G:=Group("Dic10:Dic3");

// GroupNames label

G:=SmallGroup(480,313);

// by ID

G=gap.SmallGroup(480,313);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,14118,4724]);

// Polycyclic

G:=Group<a,b,c,d|a^20=c^6=1,b^2=a^10,d^2=c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^13,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;

// generators/relations

1	0	0	0	0	0
0	240	0	0	0	0
0	0	126	0	12	12
0	0	115	127	127	115
0	0	12	12	0	126
0	0	0	229	114	229

1	0	0	0	0	0
0	240	0	0	0	0
0	0	126	0	12	12
0	0	115	127	127	115
0	0	12	12	0	126
0	0	0	229	114	229

G = Dic10⋊Dic3 order 480 = 25·3·5

1st semidirect product of Dic10 and Dic3 acting via Dic3/C3=C4

G = Dic₁₀⋊Dic₃ order 480 = 2⁵·3·5

1^st semidirect product of Dic₁₀ and Dic₃ acting via Dic₃/C₃=C₄

1	0	0	0	0	0
0	240	0	0	0	0
0	0	126	0	12	12
0	0	115	127	127	115
0	0	12	12	0	126
0	0	0	229	114	229