C15:3C8 - GroupNames

(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)

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

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

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

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

C₁₅⋊₃C₈ is a maximal subgroup of
D₅×C₃⋊C₈ S₃×C₅⋊₂C₈ C₂₀.₃₂D₆ D₆.Dic₅ C₁₅⋊D₈ C₃₀.D₄ C₂₀.D₆ C₁₅⋊Q₁₆ C₈×D₁₅ C₄₀⋊S₃ C₆₀.₇C₄ D₄⋊D₁₅ D₄.D₁₅ Q₈⋊₂D₁₅ C₁₅⋊₇Q₁₆ C₄₅⋊₃C₈ C₆₀.S₃ C₂₀.S₄ C₅⋊₂U₂(𝔽₃)
C₁₅⋊₃C₈ is a maximal quotient of
C₁₅⋊₃C₁₆ C₄₅⋊₃C₈ C₆₀.S₃ C₂₀.S₄

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+			+	+	-	-		+		-
image	C₁	C₂	C₄	C₈	S₃	D₅	Dic₃	Dic₅	C₃⋊C₈	D₁₅	C₅⋊₂C₈	Dic₁₅	C₁₅⋊₃C₈
kernel	C₁₅⋊₃C₈	C₆₀	C₃₀	C₁₅	C₂₀	C₁₂	C₁₀	C₆	C₅	C₄	C₃	C₂	C₁
# reps	1	1	2	4	1	2	1	2	2	4	4	4	8

Matrix representation of C₁₅⋊₃C₈ ►in GL₂(𝔽₂₉) generated by

1	17
28	13

0	17
1	0

G:=sub<GL(2,GF(29))| [1,28,17,13],[0,1,17,0] >;

C₁₅⋊₃C₈ in GAP, Magma, Sage, TeX

C_{15}\rtimes_3C_8

% in TeX

G:=Group("C15:3C8");

// GroupNames label

G:=SmallGroup(120,3);

// by ID

G=gap.SmallGroup(120,3);

# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,10,26,323,2404]);

// Polycyclic

G:=Group<a,b|a^15=b^8=1,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of C₁₅⋊₃C₈ in TeX

G = C15⋊3C8 order 120 = 23·3·5

1st semidirect product of C15 and C8 acting via C8/C4=C2

G = C₁₅⋊₃C₈ order 120 = 2³·3·5

1^st semidirect product of C₁₅ and C₈ acting via C₈/C₄=C₂