C₇×#7536 #13184752

order	1792
nilpotency class	6
isoclinism	isoclinic class of #7535
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₇×#7536

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1409
commutator	C₄×C₈	C₂×C₄×C₇
Frattini, cosocle	#492	C₂²×C₇
prime radical	C₇×#127	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₄²×C₇ ◁ C₇×#492 ◁ G

138 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 2 generators on 39 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 524);