C₇₁×#12863 #31833735

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₁×#12863

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₇₁
prime radical	C₂²×#41×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂²×#41×C₇₁ ◁ G

614 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 95 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,19,11,15)(12,16,20,24,21,17,22,23)(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);(2,5)(6,8)(9,11,14,13)(10,15,19,18)(12,20,21,22)(16,24,17,23);(1,3,4,7)(2,6,5,8)(10,16)(11,13)(12,21)(15,23)(17,19)(18,24)

Conjugacy classes & character table

2414 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5851);