C₂×C₇×#1308 #13185339

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇×#1308

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#95
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂⁴×C₇	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄²×C₇ ◁ G

526 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 41
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2)(35,36,37,38,39,40,41);(3,4,6,11,8,10,7,12)(5,13,14,18,15,9,16,17)(19,20,22,25,23,26,30,33)(21,27,29,34,28,24,31,32);(3,5,7,14,8,15,6,16)(4,9,12,17,10,13,11,18)(19,21)(20,24,26,27)(22,29)(23,28)(25,32,33,34)(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, 1111);