D4xC15 - 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)

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

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

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

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

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

D₄×C₁₅ is a maximal subgroup of D₄⋊D₁₅ D₄.D₁₅ D₄⋊₂D₁₅

75 conjugacy classes

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

75 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+										+
image	C₁	C₂	C₂	C₃	C₅	C₆	C₆	C₁₀	C₁₀	C₁₅	C₃₀	C₃₀	D₄	C₃×D₄	C₅×D₄	D₄×C₁₅
kernel	D₄×C₁₅	C₆₀	C₂×C₃₀	C₅×D₄	C₃×D₄	C₂₀	C₂×C₁₀	C₁₂	C₂×C₆	D₄	C₄	C₂²	C₁₅	C₅	C₃	C₁
# reps	1	1	2	2	4	2	4	4	8	8	8	16	1	2	4	8

Matrix representation of D₄×C₁₅ ►in GL₂(𝔽₃₁) generated by

19	0
0	19

0	24
9	0

1	0
0	30

G:=sub<GL(2,GF(31))| [19,0,0,19],[0,9,24,0],[1,0,0,30] >;

D₄×C₁₅ in GAP, Magma, Sage, TeX

D_4\times C_{15}

% in TeX

G:=Group("D4xC15");

// GroupNames label

G:=SmallGroup(120,32);

// by ID

G=gap.SmallGroup(120,32);

# by ID

G:=PCGroup([5,-2,-2,-3,-5,-2,621]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of D₄×C₁₅ in TeX

G = D4×C15 order 120 = 23·3·5

Direct product of C15 and D4

G = D₄×C₁₅ order 120 = 2³·3·5

Direct product of C₁₅ and D₄