ΓL(1,𝔽₈)×#1455 #72229484

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₈

Aliases

Fitting, Hall	C₇×#1455 ⋊ C₃
Sylow	C₇ ⋊ C₃×#1455

428 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 228 normal subgroups are shown.

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

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

(2,3,4)(5,7,6)(8,9,11,13,12,14,10,15);(1,2,4,6,3,5,7)(8,10,12,11)(9,13,14,15)(17,20)(21,23);(16,17,18,21,19,20,22,23)

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

maximal subgroup of	AGL(1,𝔽₇)×#1455, ΓL(1,𝔽₈)×#19745, #72259710
maximal normal subgroup of	AGL(1,𝔽₇)×#1455, ΓL(1,𝔽₈)×#19745, #72259710

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

gap> G := Group((2,3,4)(5,7,6)(8,9,11,13,12,14,10,15), (1,2,4,6,3,5,7)(8,10,12,11)(9,13,14,15)(17,20)(21,23), (16,17,18,21,19,20,22,23));