C₇×#13572 #13190788

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13572

Subgroups

	subgroup	G/H
center	C₄×C₇	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,9,12,18,14,20,28,36)(10,21,24,32,25,34,39,15)(11,16,26,33,13,19,30,17)(22,38,23,37,29,35,27,31);(8,10)(9,15)(11,22)(12,27)(13,29)(14,25)(16,31)(17,34)(18,35)(19,37)(20,32)(21,33)(23,28)(24,26)(30,39)(36,38);(8,11,14,13)(9,16,20,19)(10,22,25,29)(12,26,28,30)(15,31,32,37)(17,18,33,36)(21,38,34,35)(23,39,27,24)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6560);