C₇×#2817 #11742126

order	896
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2817

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 184 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 35
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3,4,5,6,7)(13,16)(17,19)(20,21,25,23)(22,28,30,26)(24,27,33,31)(29,35,32,34);(9,11)(13,16)(14,18)(20,22,25,30)(21,26,23,28)(24,32,33,29)(27,35,31,34);(8,9)(10,11)(12,13)(14,17)(15,16)(18,19)(20,23,25,21)(22,26,30,28)(24,27,33,31)(29,35,32,34);(12,14)(13,17)(15,18)(16,19)(20,24,25,33)(21,27,23,31)(22,29,30,32)(26,34,28,35)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13254440, #13254457, #13265854, …
maximal subgroup of	C₇×#9982, C₇×#9985, C₇×#12095, …
maximal normal subgroup of	C₇×#9982, C₇×#9985, C₇×#12095, …
maximal quotient of	C₇×#14626, C₇×#14627, C₇×#14629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16892);