C₇×#2229 #11730698

order	896
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#2229

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 124 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) = 47
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9,11,13)(10,14,15,12)(16,17,19,22)(18,26,27,44)(20,32,30,45)(21,35,31,25)(23,39,37,28)(24,41,38,34)(29,46,36,42)(33,43,47,40);(8,10,11,15)(9,12,13,14)(17,22)(23,37)(24,38)(25,35)(26,44)(32,45)(40,43)(42,46);(16,18,19,27)(17,23,22,37)(20,33,30,47)(21,36,31,29)(24,42,38,46)(25,43,35,40)(26,45,44,32)(28,41,39,34)

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#2229, #13508441, #13508457, …
maximal subgroup of	C₂×C₇×#2229, C₇×#33626, C₇×#33628, …
maximal normal subgroup of	C₂×C₇×#2229, C₇×#33626, C₇×#33628, …
maximal quotient of	C₇×#12814, C₇×#12936, C₇×#12938, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5464);