C₇×#12876 #13190092

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12876

Subgroups

	subgroup	G/H
center	C₄×C₇	#481
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1824
prime radical	C₂⁴×C₇	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×M₁₆ ◁ G

370 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5864);