C₇×#2222 #11730691

order	896
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2222

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#131 ◁ G

140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,15,21)(10,22,28,50)(11,29,32,46)(12,33,35,44)(13,20,14,19)(16,37,41,25)(17,36,43,31)(18,34,45,23)(24,53,56,70)(26,49,27,48)(30,61,62,42)(38,65,66,71)(39,57,40,51)(47,58,69,55)(52,60,63,68)(54,59,64,67);(8,10,23,51,14,27,35,37,15,28,34,57,13,26,12,25)(9,16,33,49,20,40,45,50,21,41,44,48,19,39,18,22)(11,24,52,71,31,55,62,64,32,56,63,65,36,58,30,54)(17,38,60,53,29,59,61,69,43,66,68,70,46,67,42,47);(8,11)(9,17)(10,24)(12,30)(13,36)(14,31)(15,32)(16,38)(18,42)(19,46)(20,29)(21,43)(22,47)(23,52)(25,54)(26,58)(27,55)(28,56)(33,60)(34,63)(35,62)(37,64)(39,67)(40,59)(41,66)(44,68)(45,61)(48,70)(49,53)(50,69)(51,71)(57,65)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13274492, #13274499, #13274664, …
maximal subgroup of	C₇×#13671, C₇×#13672, C₇×#13701, …
maximal normal subgroup of	C₇×#13671, C₇×#13672, C₇×#13701, …
maximal quotient of	C₇×#12431, C₇×#12434, C₇×#12440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5457);