C₇×#1720 #11730189

order	896
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₇×#1720

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 72 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)(8,9,13,10)(11,12,15,14)(16,17,20,23)(18,24,26,31)(19,28,25,22)(21,29,30,27);(8,10,12,15,13,9,14,11)(16,18,25,29,20,26,19,27)(17,21,28,31,23,30,22,24);(8,11,12,10,13,15,14,9)(18,26)(21,30)(24,31)(27,29)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1720, #13341566, #13341616, …
maximal subgroup of	C₂×C₇×#1720, C₇×#19613, C₇×#19619, …
maximal normal subgroup of	C₂×C₇×#1720, C₇×#19613, C₇×#19619, …
maximal quotient of	C₇×#8821, C₇×#9827, C₇×#9828, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4955);