C₇×#1291 #11725657

order	896
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#1291

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#95
commutator	C₂×C₄	C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂²×C₇
prime radical	C₂³×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 2 generators on 31 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#1291, #13243710, #13243714, …
maximal subgroup of	C₂×C₇×#1291, C₇×#7992, C₇×#7993, …
maximal normal subgroup of	C₂×C₇×#1291, C₇×#7992, C₇×#7993, …
maximal quotient of	C₇×#7019, C₇×#7021, C₇×#7022, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 423);