C₇×#2234 #11730703

order	896
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#2234

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₇×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

142 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2234, #13508594, #13508610, …
maximal subgroup of	C₂×C₇×#2234, C₇×#33645, C₇×#33647, …
maximal normal subgroup of	C₂×C₇×#2234, C₇×#33645, C₇×#33647, …
maximal quotient of	C₇×#12868, C₇×#12921, C₇×#12992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5469);