C₇×#3363 #11742672

order	896
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#3363

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×Q₈ ◁ G

336 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13343999, #13344059, #13344069, …
maximal subgroup of	C₇×#19812, C₇×#19816, C₇×#19817, …
maximal normal subgroup of	C₇×#19812, C₇×#19816, C₇×#19817, …
maximal quotient of	C₇×Q₈×#103, C₇×#22542, C₇×#22740, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 17438);