C₂⁴×C₃²×ΓL(1,𝔽₈) #72247012

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃³ · Schur multiplier (H₂) = C₂⁶×C₃³

Aliases

	subgroup	G/H
center, hypercenter	C₂⁴×C₃²	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂⁴×C₃³
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃²×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴×C₃²

3752 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

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

maximal subgroup of	C₂⁴×C₃⋊D₃×ΓL(1,𝔽₈), C₂⁴×C₃²×AGL(1,𝔽₇), C₂⁴×C₃⋊(C₃⋊AGL(1,𝔽₇)), …
maximal normal subgroup of	C₂⁴×C₃⋊D₃×ΓL(1,𝔽₈), C₂⁴×C₃²×AGL(1,𝔽₇), C₂⁴×C₃⋊(C₃⋊AGL(1,𝔽₇)), …

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

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