C₂⁴×C₃⋊D₃×ΓL(1,𝔽₈) #72247014

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₂⁴	C₃⋊D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃²×C₇	C₂⁵×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃²×C₇	C₂×C₃
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂⁴

10048 conjugacy classes of subgroups; only the 2127 normal subgroups are shown.

normal subgroups only

This group has 10048 conjugacy classes of subgroups — too many to draw. Only its 2127 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 1008

A minimal faithful permutation representation — 5 generators on 21 points:

(7,8)(9,10,11)(16,17,18)(19,21,20);(5,6)(12,13,14)(15,16,18,20,17,19,21);(3,4);(10,11)(13,14);(1,2)

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

maximal subgroup of	C₂⁴×C₃⋊D₃×AGL(1,𝔽₇), C₃⋊D₃×ΓL(1,𝔽₈)×#120, C₃⋊D₃×#10620789
maximal normal subgroup of	C₂⁴×C₃⋊D₃×AGL(1,𝔽₇), C₃⋊D₃×ΓL(1,𝔽₈)×#120, C₃⋊D₃×#10620789

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((7,8)(9,10,11)(16,17,18)(19,21,20), (5,6)(12,13,14)(15,16,18,20,17,19,21), (3,4), (10,11)(13,14), (1,2));