C₇×#1532 #11730001

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

Aliases

direct product

C₇×#1532

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₇×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 31 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13245315, #13245320, #13246182, …
maximal subgroup of	C₇×#8348, C₇×#8349, C₇×#8497, …
maximal normal subgroup of	C₇×#8348, C₇×#8349, C₇×#8497, …
maximal quotient of	C₇×#7580, C₇×#7581, C₇×#7586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4767);