C5xC3:D12 - GroupNames

G = C₅×C₃⋊D₁₂ order 360 = 2³·3²·5

Direct product of C₅ and C₃⋊D₁₂

direct product, metabelian, supersoluble, monomial

Aliases: C₅×C₃⋊D₁₂, C₁₅⋊₉D₁₂, C₃₀.₃₈D₆, C₁₀.₁₆S₃², D₆⋊₂(C₅×S₃), C₃⋊₂(C₅×D₁₂), (C₃×C₁₅)⋊₁₆D₄, Dic₃⋊(C₅×S₃), (S₃×C₃₀)⋊₈C₂, (S₃×C₆)⋊₂C₁₀, (S₃×C₁₀)⋊₅S₃, C₃²⋊₃(C₅×D₄), C₆.₄(S₃×C₁₀), C₁₅⋊₉(C₃⋊D₄), (C₅×Dic₃)⋊₄S₃, (C₃×Dic₃)⋊₁C₁₀, (Dic₃×C₁₅)⋊₅C₂, (C₃×C₃₀).₃₀C₂², C₂.₄(C₅×S₃²), C₃⋊₁(C₅×C₃⋊D₄), (C₁₀×C₃⋊S₃)⋊₆C₂, (C₂×C₃⋊S₃)⋊₁C₁₀, (C₃×C₆).₄(C₂×C₁₀), SmallGroup(360,75)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₅×C₃⋊D₁₂

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₃₀ — S₃×C₃₀ — C₅×C₃⋊D₁₂

Lower central

C₃² — C₃×C₆ — C₅×C₃⋊D₁₂

Upper central

C₁ — C₁₀

Generators and relations for C₅×C₃⋊D₁₂
G = < a,b,c,d | a⁵=b³=c¹²=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 252 in 74 conjugacy classes, 28 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², C₁₀, C₁₀, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₁₅, C₁₅, C₃×S₃, C₃⋊S₃, C₃×C₆, C₂₀, C₂×C₁₀, D₁₂, C₃⋊D₄, C₅×S₃, C₃₀, C₃₀, C₃×Dic₃, S₃×C₆, C₂×C₃⋊S₃, C₅×D₄, C₃×C₁₅, C₅×Dic₃, C₆₀, S₃×C₁₀, S₃×C₁₀, C₂×C₃₀, C₃⋊D₁₂, S₃×C₁₅, C₅×C₃⋊S₃, C₃×C₃₀, C₅×D₁₂, C₅×C₃⋊D₄, Dic₃×C₁₅, S₃×C₃₀, C₁₀×C₃⋊S₃, C₅×C₃⋊D₁₂
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₁₀, D₆, C₂×C₁₀, D₁₂, C₃⋊D₄, C₅×S₃, S₃², C₅×D₄, S₃×C₁₀, C₃⋊D₁₂, C₅×D₁₂, C₅×C₃⋊D₄, C₅×S₃², C₅×C₃⋊D₁₂

Smallest permutation representation of C₅×C₃⋊D₁₂
►On 60 points

Generators in S₆₀

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

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

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

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

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

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

75 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4	5A	5B	5C	5D	6A	6B	6C	6D	6E	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	10K	10L	12A	12B	15A	···	15H	15I	15J	15K	15L	20A	20B	20C	20D	30A	···	30H	30I	30J	30K	30L	30M	···	30T	60A	···	60H
order	1	2	2	2	3	3	3	4	5	5	5	5	6	6	6	6	6	10	10	10	10	10	10	10	10	10	10	10	10	12	12	15	···	15	15	15	15	15	20	20	20	20	30	···	30	30	30	30	30	30	···	30	60	···	60
size	1	1	6	18	2	2	4	6	1	1	1	1	2	2	4	6	6	1	1	1	1	6	6	6	6	18	18	18	18	6	6	2	···	2	4	4	4	4	6	6	6	6	2	···	2	4	4	4	4	6	···	6	6	···	6

75 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+					+	+	+	+	+								+	+
image	C₁	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	S₃	S₃	D₄	D₆	D₁₂	C₃⋊D₄	C₅×S₃	C₅×S₃	C₅×D₄	S₃×C₁₀	C₅×D₁₂	C₅×C₃⋊D₄	S₃²	C₃⋊D₁₂	C₅×S₃²	C₅×C₃⋊D₁₂
kernel	C₅×C₃⋊D₁₂	Dic₃×C₁₅	S₃×C₃₀	C₁₀×C₃⋊S₃	C₃⋊D₁₂	C₃×Dic₃	S₃×C₆	C₂×C₃⋊S₃	C₅×Dic₃	S₃×C₁₀	C₃×C₁₅	C₃₀	C₁₅	C₁₅	Dic₃	D₆	C₃²	C₆	C₃	C₃	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	4	4	4	4	1	1	1	2	2	2	4	4	4	8	8	8	1	1	4	4

Matrix representation of C₅×C₃⋊D₁₂ ►in GL₆(𝔽₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	58	0	0	0
0	0	0	58	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	60
0	0	0	0	1	60

0	1	0	0	0	0
60	0	0	0	0	0
0	0	1	1	0	0
0	0	60	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

60	0	0	0	0	0
0	1	0	0	0	0
0	0	60	60	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₅×C₃⋊D₁₂ in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes D_{12}

% in TeX

G:=Group("C5xC3:D12");

// GroupNames label

G:=SmallGroup(360,75);

// by ID

G=gap.SmallGroup(360,75);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,1210,8645]);

// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C5×C3⋊D12 order 360 = 23·32·5

Direct product of C5 and C3⋊D12

G = C₅×C₃⋊D₁₂ order 360 = 2³·3²·5

Direct product of C₅ and C₃⋊D₁₂