C₇×#1977 #11730446

order	896
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#1977

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

296 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1977, #13357798, #13357806, …
maximal subgroup of	C₂×C₇×#1977, C₇×#21248, C₇×#21249, …
maximal normal subgroup of	C₂×C₇×#1977, C₇×#21248, C₇×#21249, …
maximal quotient of	C₇×#9667, C₇×#9672, C₇×#9723, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5212);