C60.99D4 - GroupNames

G = C₆₀.₉₉D₄ order 480 = 2⁵·3·5

99^th non-split extension by C₆₀ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆₀.₉₉D₄, D₁₂⋊₂Dic₅, C₂₀.₅₆D₁₂, Dic₆⋊₂Dic₅, C₁₅⋊₈C₄≀C₂, (C₅×D₁₂)⋊₈C₄, C₂₀.₄₀(C₄×S₃), (C₅×Dic₆)⋊₈C₄, (C₄×Dic₅)⋊₁S₃, C₄○D₁₂.₁D₅, (C₂×C₂₀).₅₆D₆, (C₂×C₃₀).₂₆D₄, C₄.₃(S₃×Dic₅), C₅⋊₅(C₄²⋊₄S₃), C₆₀.₁₀₉(C₂×C₄), (C₁₂×Dic₅)⋊₁C₂, C₆₀.₇C₄⋊₁₁C₂, C₁₀.₄₆(D₆⋊C₄), (C₂×C₁₂).₃₁₂D₁₀, C₃⋊₂(D₄⋊₂Dic₅), C₁₂.₆₄(C₅⋊D₄), C₄.₂₈(C₅⋊D₁₂), (C₂×C₆₀).₄₃C₂², C₁₂.₁₈(C₂×Dic₅), C₆.₉(C₂³.D₅), C₃₀.₆₂(C₂²⋊C₄), C₂.₁₀(D₆⋊Dic₅), C₂².₂(C₁₅⋊D₄), (C₅×C₄○D₁₂).₂C₂, (C₂×C₄).₁₃₇(S₃×D₅), (C₂×C₁₀).₂(C₃⋊D₄), (C₂×C₆).₄₇(C₅⋊D₄), SmallGroup(480,55)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — C₆₀.₉₉D₄

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — C₂×C₆₀ — C₁₂×Dic₅ — C₆₀.₉₉D₄

Lower central

C₁₅ — C₃₀ — C₆₀ — C₆₀.₉₉D₄

Upper central

C₁ — C₄ — C₂×C₄

Generators and relations for C₆₀.₉₉D₄
G = < a,b,c | a⁶⁰=b⁴=1, c²=a¹⁵, bab^-1=a⁴⁹, cac^-1=a²⁹, cbc^-1=a¹⁵b^-1 >

Subgroups: 332 in 88 conjugacy classes, 34 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₁₅, C₄², M₄(2), C₄○D₄, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₃⋊C₈, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₅×S₃, C₃₀, C₃₀, C₄≀C₂, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₄.Dic₃, C₄×C₁₂, C₄○D₁₂, C₅×Dic₃, C₃×Dic₅, C₆₀, S₃×C₁₀, C₂×C₃₀, C₄.Dic₅, C₄×Dic₅, C₅×C₄○D₄, C₄²⋊₄S₃, C₁₅⋊₃C₈, C₆×Dic₅, C₅×Dic₆, S₃×C₂₀, C₅×D₁₂, C₅×C₃⋊D₄, C₂×C₆₀, D₄⋊₂Dic₅, C₁₂×Dic₅, C₆₀.₇C₄, C₅×C₄○D₁₂, C₆₀.₉₉D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, D₅, D₆, C₂²⋊C₄, Dic₅, D₁₀, C₄×S₃, D₁₂, C₃⋊D₄, C₄≀C₂, C₂×Dic₅, C₅⋊D₄, D₆⋊C₄, S₃×D₅, C₂³.D₅, C₄²⋊₄S₃, S₃×Dic₅, C₁₅⋊D₄, C₅⋊D₁₂, D₄⋊₂Dic₅, D₆⋊Dic₅, C₆₀.₉₉D₄

Smallest permutation representation of C₆₀.₉₉D₄
►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 31)(2 20)(3 9)(4 58)(5 47)(6 36)(7 25)(8 14)(10 52)(11 41)(12 30)(13 19)(15 57)(16 46)(17 35)(18 24)(21 51)(22 40)(23 29)(26 56)(27 45)(28 34)(32 50)(33 39)(37 55)(38 44)(42 60)(43 49)(48 54)(53 59)(61 94 91 64)(62 83 92 113)(63 72 93 102)(65 110 95 80)(66 99 96 69)(67 88 97 118)(68 77 98 107)(70 115 100 85)(71 104 101 74)(73 82 103 112)(75 120 105 90)(76 109 106 79)(78 87 108 117)(81 114 111 84)(86 119 116 89)

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+			+	+	+	+	+	-	-	+		+						+	-	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₄	D₅	D₆	Dic₅	Dic₅	D₁₀	C₄×S₃	D₁₂	C₃⋊D₄	C₄≀C₂	C₅⋊D₄	C₅⋊D₄	C₄²⋊₄S₃	S₃×D₅	S₃×Dic₅	C₅⋊D₁₂	C₁₅⋊D₄	D₄⋊₂Dic₅	C₆₀.₉₉D₄
kernel	C₆₀.₉₉D₄	C₁₂×Dic₅	C₆₀.₇C₄	C₅×C₄○D₁₂	C₅×Dic₆	C₅×D₁₂	C₄×Dic₅	C₆₀	C₂×C₃₀	C₄○D₁₂	C₂×C₂₀	Dic₆	D₁₂	C₂×C₁₂	C₂₀	C₂₀	C₂×C₁₀	C₁₅	C₁₂	C₂×C₆	C₅	C₂×C₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	1	2	2	1	1	1	2	1	2	2	2	2	2	2	4	4	4	8	2	2	2	2	4	8

Matrix representation of C₆₀.₉₉D₄ ►in GL₆(𝔽₂₄₁)

16	0	0	0	0	0
0	226	0	0	0	0
0	0	64	0	0	0
0	0	0	64	0	0
0	0	0	0	1	51
0	0	0	0	190	51

1	0	0	0	0	0
0	240	0	0	0	0
0	0	240	0	0	0
0	0	0	64	0	0
0	0	0	0	1	51
0	0	0	0	0	240

0	240	0	0	0	0
1	0	0	0	0	0
0	0	0	64	0	0
0	0	240	0	0	0
0	0	0	0	165	172
0	0	0	0	192	76

G:=sub<GL(6,GF(241))| [16,0,0,0,0,0,0,226,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,190,0,0,0,0,51,51],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,51,240],[0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,64,0,0,0,0,0,0,0,165,192,0,0,0,0,172,76] >;

C₆₀.₉₉D₄ in GAP, Magma, Sage, TeX

C_{60}._{99}D_4

% in TeX

G:=Group("C60.99D4");

// GroupNames label

G:=SmallGroup(480,55);

// by ID

G=gap.SmallGroup(480,55);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,346,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^15,b*a*b^-1=a^49,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;

// generators/relations

G = C60.99D4 order 480 = 25·3·5

99th non-split extension by C60 of D4 acting via D4/C2=C22

G = C₆₀.₉₉D₄ order 480 = 2⁵·3·5

99^th non-split extension by C₆₀ of D₄ acting via D₄/C₂=C₂²