GL(2,𝔽₃)×AGL(1,𝔽₁₃) #72259549

order	7488
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃×SL(2,𝔽₃)	C₂×C₃×C₄
Fitting	Q₈×C₁₃	C₃×C₄×D₃
socle	C₂×C₁₃	C₃×C₄×S₄
prime radical	GL(2,𝔽₃)×D₁₃⋊C₃	C₂
cosocle	SL(2,𝔽₃)×D₁₃	C₂²×C₃
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

374 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

maximal subgroup of	AGL(1,𝔽₁₃)×#5935
automorphism group of	C₃²×D₁₃

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

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