C₇×#13054 #13190270

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13054

Subgroups

	subgroup	G/H
center	C₄×C₇	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₂×#41	C₂³×C₇
socle	C₂×C₇	#1889
prime radical	C₇×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

632 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9,13,18)(10,19)(11,16,20,29)(12,17)(14,24)(15,27)(21,34)(22,31)(23,36,26,39)(25,33)(28,35)(30,37,32,38);(8,10,20,22,13,21,11,24)(9,14,29,19,18,31,16,34)(12,26,28,38,27,23,25,37)(15,32,33,39,17,30,35,36);(9,15)(10,21)(12,27)(14,30)(16,35)(17,18)(19,36)(22,24)(25,28)(29,33)(31,32)(34,39)

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, 6042);