C₃×C₄×C₄₁×C₆₁ #44473103

order	30012
nilpotency class	1
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₀₀₁₂
abelian	C₃×C₄×C₄₁×C₆₁
multiplicative	(ℤ/30013)ˣ, (ℤ/60026)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃×C₄₁×C₆₁
socle, prime radical	C₂×C₃×C₄₁×C₆₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 30012

A minimal faithful permutation representation — 1 generator on 109 points:

(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)

Conjugacy classes & character table

30012 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Dic₁₅₀₀₆, D₃₀₀₁₂
maximal subgroup of	Dic₁₅₀₀₆, D₃₀₀₁₂
maximal normal subgroup of	Dic₁₅₀₀₆, D₃₀₀₁₂
automorphism group of	C₃₀₀₁₃
outer automorphism group of	C₃₀₀₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30012, 1);

C3×C4×C41×C61 #44473103