C₇×#2852 #11742161

order	896
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2852

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

582 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 27
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13526479, #13526510, #13526544, …
maximal subgroup of	C₇×#34990, C₇×#34991, C₇×#34995, …
maximal normal subgroup of	C₇×#34990, C₇×#34991, C₇×#34995, …
maximal quotient of	C₇×#14683, C₇×#14695, C₇×#14696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16927);