C₇×#1506 #11729975

order	896
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1506

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 228 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)(9,12,13,14)(16,17)(18,24,26,21)(19,22)(20,23)(25,30,27,29)(28,31);(8,9)(10,14)(11,13)(12,15)(16,18,20,26)(17,21,23,24)(19,27,28,25)(22,30,31,29);(16,19,20,28)(17,22,23,31)(18,25,26,27)(21,29,24,30)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13245656, #13245670, #13246153, …
maximal subgroup of	C₇×#8413, C₇×#8415, C₇×#8492, …
maximal normal subgroup of	C₇×#8413, C₇×#8415, C₇×#8492, …
maximal quotient of	C₇×#7581, C₇×#7588, C₇×#7590, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4741);