C₇×#1469 #11729938

order	896
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1469

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	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

256 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 224

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13246045, #13246055, #13246097, …
maximal subgroup of	C₇×#8475, C₇×#8477, C₇×#8483, …
maximal normal subgroup of	C₇×#8475, C₇×#8477, C₇×#8483, …
maximal quotient of	C₇×#7656, C₇×#7659, C₇×#7662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4704);