C₇×#3080 #11742389

order	896
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#3080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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)(8,9,11,13)(10,14,15,12)(16,17,20,22)(18,26,28,31)(19,25,21,24)(23,29,30,27);(8,10,11,15)(9,12,13,14)(17,22)(23,30)(24,25)(26,31);(16,18)(17,23)(19,27)(20,28)(21,29)(22,30)(24,26)(25,31);(16,19,20,21)(17,24,22,25)(18,27,28,29)(23,26,30,31)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13250889, #13250896, #13395299, …
maximal subgroup of	C₇×#9351, C₇×#9352, C₇×#24724, …
maximal normal subgroup of	C₇×#9351, C₇×#9352, C₇×#24724, …
maximal quotient of	C₇×#20649, C₇×#20875, C₇×#21034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 17155);