C₇×#2853 #11742162

order	896
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#2853

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)(13,15)(14,18)(21,24)(22,26);(12,13,16,15)(14,17,18,19)(20,21)(22,25)(23,24)(26,27);(12,14)(13,17)(15,19)(16,18);(8,9)(10,11)(12,15,16,13)(14,19,18,17)(20,22,23,26)(21,25,24,27)

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13387036, #13387660, #13526362, …
maximal subgroup of	C₇×#24057, C₇×#24113, C₇×#34983, …
maximal normal subgroup of	C₇×#24057, C₇×#24113, C₇×#34983, …
maximal quotient of	C₇×#14684, C₇×#14695, C₇×#14696, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16928);