C₇₃×#7166 #32916570

order	18688
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₃×#7166

Subgroups

	subgroup	G/H
center	C₄²×C₇₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₇₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₃
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

216 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 97
transitive degree	μ_t(G) = 9344

A minimal faithful permutation representation — 2 generators on 97 points:

(1,2)(3,5)(4,7)(6,8)(9,10,12,18,15,11,16,20,14,17,13,23,22,19,24,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,5,7,6)(9,11,13,20,15,17,24,23,14,19,12,21,22,10,16,18)

Conjugacy classes & character table

5548 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18688, 154);