C₇₁×#7123 #31827995

order	18176
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₁×#7123

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₁	#99
commutator	C₂²×C₄	C₂×C₈×C₇₁
Frattini, cosocle	C₂×#95	C₂²×C₇₁
prime radical	C₂⁴×C₇₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂⁴×C₇₁ ◁ C₂×#95×C₇₁ ◁ G

412 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 4544

A minimal faithful permutation representation — 2 generators on 95 points:

(2,5,6,7)(9,10,11,13,12,14,15,16)(17,18)(19,22,24,21)(20,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(1,2)(3,7)(4,6)(5,8)(18,21,23,22)

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 111);