C₇×#19616 #13196832

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#19616

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

906 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 278 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) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 12604);