C₇×#2659 #11741968

order	896
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#2659

Subgroups

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

524 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 448

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13254350, #13254355, #13254384, …
maximal subgroup of	C₇×#9966, C₇×#9967, C₇×#9972, …
maximal normal subgroup of	C₇×#9966, C₇×#9967, C₇×#9972, …
maximal quotient of	C₇×#14275, C₇×#14281, C₇×#14327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 16734);