C₂⁴×D₇×C₃⋊D₃ #72246714

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₃

Aliases

C₂⁴×D₇×C₃⋊D₃

	subgroup	G/H
center, hypercenter	C₂⁴	D₇×C₃⋊D₃
commutator, cosocle, hypocenter	C₃²×C₇	C₂⁶
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃²×C₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂⁴

33900 conjugacy classes of subgroups; only the 5404 normal subgroups are shown.

normal subgroups only

This group has 33900 conjugacy classes of subgroups — too many to draw. Only its 5404 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 — 6 generators on 21 points:

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

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

maximal subgroup of	D₇×C₃⋊D₃×#120, C₂⁴×C₃⋊D₃×AGL(1,𝔽₇)
maximal normal subgroup of	D₇×C₃⋊D₃×#120, C₂⁴×C₃⋊D₃×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)(15,16,18,20,21,19,17), (5,6)(12,13,14), (3,4), (1,2)(3,4)(5,6)(7,8)(10,11)(13,14), (16,17)(18,19)(20,21), (1,2));