C30.4C4^2 - GroupNames

G = C₃₀.₄C₄² order 480 = 2⁵·3·5

4^th non-split extension by C₃₀ of C₄² acting via C₄²/C₂=C₂xC₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₃₀.₄C₄², C₃:C₈:₄F₅, C₅:C₈:₁Dic₃, C₁₅:₃C₈:₆C₄, C₆.₉(C₄xF₅), D₅:C₈.₂S₃, C₃:₂(C₈:F₅), C₅:₁(C₂₄:C₄), C₁₅:₂(C₈:C₄), C₄.₂₅(S₃xF₅), C₂₀.₂₅(C₄xS₃), C₆₀.₂₅(C₂xC₄), (C₄xD₅).₇₁D₆, D₁₀.₈(C₄xS₃), C₁₂.₃₂(C₂xF₅), C₂.₅(Dic₃xF₅), C₁₀.₄(C₄xDic₃), D₅.₁(C₈:S₃), (C₃xD₅).₂M₄(2), Dic₅.₈(C₂xDic₃), (D₅xC₁₂).₆₃C₂², (C₅xC₃:C₈):₆C₄, (C₃xC₅:C₈):₂C₄, (D₅xC₃:C₈).₉C₂, (C₄xC₃:F₅).₃C₂, (C₂xC₃:F₅).₂C₄, (C₃xD₅:C₈).₃C₂, (C₆xD₅).₁₂(C₂xC₄), (C₃xDic₅).₁₆(C₂xC₄), SmallGroup(480,226)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₃₀.₄C₄²

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xDic₅ — D₅xC₁₂ — C₃xD₅:C₈ — C₃₀.₄C₄²

Lower central

C₁₅ — C₃₀ — C₃₀.₄C₄²

Upper central

C₁ — C₄

Generators and relations for C₃₀.₄C₄²
G = < a,b,c | a³⁰=1, b⁴=c⁴=a¹⁵, bab^-1=a¹³, cac^-1=a¹¹, cbc^-1=a¹⁵b >

Subgroups: 356 in 80 conjugacy classes, 36 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, C₆, C₆, C₈, C₂xC₄, D₅, C₁₀, Dic₃, C₁₂, C₁₂, C₂xC₆, C₁₅, C₄², C₂xC₈, Dic₅, C₂₀, F₅, D₁₀, C₃:C₈, C₃:C₈, C₂₄, C₂xDic₃, C₂xC₁₂, C₃xD₅, C₃₀, C₈:C₄, C₅:₂C₈, C₄₀, C₅:C₈, C₄xD₅, C₂xF₅, C₂xC₃:C₈, C₄xDic₃, C₂xC₂₄, C₃xDic₅, C₆₀, C₃:F₅, C₆xD₅, C₈xD₅, D₅:C₈, C₄xF₅, C₂₄:C₄, C₅xC₃:C₈, C₁₅:₃C₈, C₃xC₅:C₈, D₅xC₁₂, C₂xC₃:F₅, C₈:F₅, D₅xC₃:C₈, C₃xD₅:C₈, C₄xC₃:F₅, C₃₀.₄C₄²
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, Dic₃, D₆, C₄², M₄(2), F₅, C₄xS₃, C₂xDic₃, C₈:C₄, C₂xF₅, C₈:S₃, C₄xDic₃, C₄xF₅, C₂₄:C₄, S₃xF₅, C₈:F₅, Dic₃xF₅, C₃₀.₄C₄²

Smallest permutation representation of C₃₀.₄C₄²
►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 68 36 106 16 83 51 91)(2 75 55 119 17 90 40 104)(3 82 44 102 18 67 59 117)(4 89 33 115 19 74 48 100)(5 66 52 98 20 81 37 113)(6 73 41 111 21 88 56 96)(7 80 60 94 22 65 45 109)(8 87 49 107 23 72 34 92)(9 64 38 120 24 79 53 105)(10 71 57 103 25 86 42 118)(11 78 46 116 26 63 31 101)(12 85 35 99 27 70 50 114)(13 62 54 112 28 77 39 97)(14 69 43 95 29 84 58 110)(15 76 32 108 30 61 47 93)

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	5	6A	6B	6C	8A	8B	8C	8D	8E	8F	8G	8H	10	12A	12B	12C	12D	15	20A	20B	24A	···	24H	30	40A	40B	40C	40D	60A	60B
order	1	2	2	2	3	4	4	4	4	4	4	4	4	5	6	6	6	8	8	8	8	8	8	8	8	10	12	12	12	12	15	20	20	24	···	24	30	40	40	40	40	60	60
size	1	1	5	5	2	1	1	5	5	30	30	30	30	4	2	10	10	6	6	10	10	10	10	30	30	4	2	2	10	10	8	4	4	10	···	10	8	12	12	12	12	8	8

48 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	8	8	8
type	+	+	+	+					+	-	+					+	+			+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	S₃	Dic₃	D₆	M₄(2)	C₄xS₃	C₄xS₃	C₈:S₃	F₅	C₂xF₅	C₄xF₅	C₈:F₅	S₃xF₅	Dic₃xF₅	C₃₀.₄C₄²
kernel	C₃₀.₄C₄²	D₅xC₃:C₈	C₃xD₅:C₈	C₄xC₃:F₅	C₅xC₃:C₈	C₁₅:₃C₈	C₃xC₅:C₈	C₂xC₃:F₅	D₅:C₈	C₅:C₈	C₄xD₅	C₃xD₅	C₂₀	D₁₀	D₅	C₃:C₈	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	1	1	1	2	2	4	4	1	2	1	4	2	2	8	1	1	2	4	1	1	2

Matrix representation of C₃₀.₄C₄² ►in GL₆(F₂₄₁)

240	240	0	0	0	0
1	0	0	0	0	0
0	0	0	240	0	0
0	0	0	0	240	0
0	0	0	0	0	240
0	0	1	1	1	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	213	0	185	185
0	0	185	185	0	213
0	0	56	28	56	0
0	0	28	213	213	28

64	0	0	0	0	0
177	177	0	0	0	0
0	0	136	0	31	31
0	0	210	105	210	0
0	0	0	210	105	210
0	0	31	31	0	136

G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,240,0,0,1,0,0,0,240,0,1,0,0,0,0,240,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,213,185,56,28,0,0,0,185,28,213,0,0,185,0,56,213,0,0,185,213,0,28],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,136,210,0,31,0,0,0,105,210,31,0,0,31,210,105,0,0,0,31,0,210,136] >;

C₃₀.₄C₄² in GAP, Magma, Sage, TeX

C_{30}._4C_4^2

% in TeX

G:=Group("C30.4C4^2");

// GroupNames label

G:=SmallGroup(480,226);

// by ID

G=gap.SmallGroup(480,226);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,80,1356,9414,4724]);

// Polycyclic

G:=Group<a,b,c|a^30=1,b^4=c^4=a^15,b*a*b^-1=a^13,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;

// generators/relations

64	0	0	0	0	0
177	177	0	0	0	0
0	0	136	0	31	31
0	0	210	105	210	0
0	0	0	210	105	210
0	0	31	31	0	136

64	0	0	0	0	0
177	177	0	0	0	0
0	0	136	0	31	31
0	0	210	105	210	0
0	0	0	210	105	210
0	0	31	31	0	136

G = C30.4C42 order 480 = 25·3·5

4th non-split extension by C30 of C42 acting via C42/C2=C2xC4

G = C₃₀.₄C₄² order 480 = 2⁵·3·5

4^th non-split extension by C₃₀ of C₄² acting via C₄²/C₂=C₂xC₄

64	0	0	0	0	0
177	177	0	0	0	0
0	0	136	0	31	31
0	0	210	105	210	0
0	0	0	210	105	210
0	0	31	31	0	136