C₇×#7481 #13184697

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7481
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7481

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1390
commutator	C₂×Q₈	C₄²×C₇
Frattini, cosocle	C₂×#124	C₂²×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#124 ◁ G

284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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)(10,18,20,34)(11,23,24,33)(12,26,28,17)(13,16)(14,19,29,37)(15,25)(21,36,22,35)(27,32)(30,38,31,39);(8,10)(9,14,16,30)(11,19)(12,20)(13,22)(15,18,32,35)(17,29,26,31)(21,28)(23,34,33,36)(24,38)(25,39)(27,37)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 469);