C6xD15 - GroupNames

(1 27 6 17 11 22)(2 28 7 18 12 23)(3 29 8 19 13 24)(4 30 9 20 14 25)(5 16 10 21 15 26)(31 49 41 59 36 54)(32 50 42 60 37 55)(33 51 43 46 38 56)(34 52 44 47 39 57)(35 53 45 48 40 58)

(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)

(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 45)(29 44)(30 43)

G:=sub<Sym(60)| (1,27,6,17,11,22)(2,28,7,18,12,23)(3,29,8,19,13,24)(4,30,9,20,14,25)(5,16,10,21,15,26)(31,49,41,59,36,54)(32,50,42,60,37,55)(33,51,43,46,38,56)(34,52,44,47,39,57)(35,53,45,48,40,58), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43)>;

G:=Group( (1,27,6,17,11,22)(2,28,7,18,12,23)(3,29,8,19,13,24)(4,30,9,20,14,25)(5,16,10,21,15,26)(31,49,41,59,36,54)(32,50,42,60,37,55)(33,51,43,46,38,56)(34,52,44,47,39,57)(35,53,45,48,40,58), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43) );

G=PermutationGroup([[(1,27,6,17,11,22),(2,28,7,18,12,23),(3,29,8,19,13,24),(4,30,9,20,14,25),(5,16,10,21,15,26),(31,49,41,59,36,54),(32,50,42,60,37,55),(33,51,43,46,38,56),(34,52,44,47,39,57),(35,53,45,48,40,58)], [(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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,45),(29,44),(30,43)]])

C₆×D₁₅ is a maximal subgroup of D₆⋊D₁₅ C₃⋊D₆₀ D₃₀.S₃ D₃₀⋊S₃ C₃²⋊₃D₂₀ S₃×C₆×D₅

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	5A	5B	6A	6B	6C	6D	6E	6F	6G	6H	6I	10A	10B	15A	···	15P	30A	···	30P
order	1	2	2	2	3	3	3	3	3	5	5	6	6	6	6	6	6	6	6	6	10	10	15	···	15	30	···	30
size	1	1	15	15	1	1	2	2	2	2	2	1	1	2	2	2	15	15	15	15	2	2	2	···	2	2	···	2

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+				+	+	+		+		+			+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₅	D₆	C₃×S₃	D₁₀	C₃×D₅	D₁₅	S₃×C₆	C₆×D₅	D₃₀	C₃×D₁₅	C₆×D₁₅
kernel	C₆×D₁₅	C₃×D₁₅	C₃×C₃₀	D₃₀	D₁₅	C₃₀	C₃₀	C₃×C₆	C₁₅	C₁₀	C₃²	C₆	C₆	C₅	C₃	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	2	1	2	2	4	4	2	4	4	8	8

Matrix representation of C₆×D₁₅ ►in GL₂(𝔽₃₁) generated by

6	0
0	6

28	0
22	10

4	8
2	27

G:=sub<GL(2,GF(31))| [6,0,0,6],[28,22,0,10],[4,2,8,27] >;

C₆×D₁₅ in GAP, Magma, Sage, TeX

C_6\times D_{15}

% in TeX

G:=Group("C6xD15");

// GroupNames label

G:=SmallGroup(180,34);

// by ID

G=gap.SmallGroup(180,34);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,483,3604]);

// Polycyclic

G:=Group<a,b,c|a^6=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₆×D₁₅ in TeX

G = C6×D15 order 180 = 22·32·5

Direct product of C6 and D15

G = C₆×D₁₅ order 180 = 2²·3²·5

Direct product of C₆ and D₁₅