C₇×#2825 #11742134

order	896
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₇×#2825

Subgroups

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

522 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 168 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 — 4 generators on 31 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2825, #13560191, #13560233, …
maximal subgroup of	C₂×C₇×#2825, C₇×#36670, C₇×#36676, …
maximal normal subgroup of	C₂×C₇×#2825, C₇×#36670, C₇×#36676, …
maximal quotient of	C₇×#14621, C₇×#14637, C₇×#14642, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16900);