C₇×#9485 #13186701

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9467
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#9485

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#2535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

626 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 31
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(9,12)(10,13)(14,15)(16,17,20,24,21,23,19,29)(18,25,28,30,27,31,26,22);(8,9,13,15,11,14,10,12)(16,18,21,27)(17,22)(19,26,20,28)(23,30)(24,25)(29,31);(17,23)(19,20)(22,30)(26,28)

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2473);