C₂×C₇×#2865 #13213852

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×C₇×#2865

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

2186 conjugacy classes of subgroups; only the 990 normal subgroups are shown.

normal subgroups only

This group has 2186 conjugacy classes of subgroups — too many to draw. Only its 990 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) = 896

A minimal faithful permutation representation — 5 generators on 33 points:

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

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 29624);