C₄₇×#450 #15688705

order	3008
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄₇×#450

Subgroups

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₄₇
prime radical	C₂²×D₄×C₄₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 752

A minimal faithful permutation representation — 3 generators on 59 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)(52,53,55,58)(54,56,59,57);(53,56)(54,59)(57,58);(49,51)

Conjugacy classes & character table

893 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄₇×#1596, C₄₇×#1597, C₄₇×#1599, …
maximal normal subgroup of	C₄₇×#1596, C₄₇×#1597, C₄₇×#1599, …
maximal quotient of	C₄₇×#1601, C₄₇×#1604, C₄₇×#1613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3008, 131);