C₇×#1967 #11730436

order	896
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#1967

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 312 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)(12,13,16,19,15,18,22,24)(14,17,21,26,20,25,23,27);(12,14,16,23,15,20,22,21)(13,17,24,26,18,25,19,27);(8,9,10,11)(13,18)(17,25)(19,24)(26,27)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13246093, #13246100, #13257754, …
maximal subgroup of	C₇×#8483, C₇×#8484, C₇×#10496, …
maximal normal subgroup of	C₇×#8483, C₇×#8484, C₇×#10496, …
maximal quotient of	C₇×#9651, C₇×#9652, C₇×#9687, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5202);