C₇×#2607 #11741916

order	896
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#2607

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

444 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13257025, #13257030, #13266002, …
maximal subgroup of	C₇×#10386, C₇×#10387, C₇×#12121, …
maximal normal subgroup of	C₇×#10386, C₇×#10387, C₇×#12121, …
maximal quotient of	C₇×#13953, C₇×#14091, C₇×#14093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16682);