C₂×C₇×#1324 #13188661

order	1792
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₇×#1324

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#99
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂⁴×C₇ ◁ C₂²×C₇×C₄⋊C₄ ◁ G

510 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 33
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2)(27,28,29,30,31,32,33);(3,4,5,7,6,8,9,10)(11,12)(13,19,21,25)(14,22,23,17)(15,18)(16,20)(24,26);(11,13)(12,16,18,24)(14,20)(15,21)(17,19,22,25)(23,26)

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, 4433);