C₇×#12224 #13189440

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#12224

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇ ◁ C₂×C₇×C₄⋊C₄ ◁ G

380 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,13,19)(10,22,25,35)(11,27,29,38)(12,31,23,17)(14,21,34,20)(15,30,37,24)(16,33,36,28)(18,32,39,26);(8,10,13,25)(9,15,19,37)(11,28,29,33)(12,30,23,24)(14,26,34,32)(16,39,36,18)(17,35,31,22)(20,27,21,38);(8,11,14,30)(9,16,21,17)(10,23,26,28)(12,32,33,25)(13,29,34,24)(15,35,38,39)(18,37,22,27)(19,36,20,31)

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, 5212);