C₇×#942 #10621339

order	672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#942

Splittings

Fitting	C₂×C₃×C₇×Q₈ ⋊ C₂
Hall	C₃×C₇ ⋊ #137
Sylow	C₃ ⋊ C₇×#137

Subgroups

	subgroup	G/H
center	C₂×C₇	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×Q₈	C₂
socle	C₂×C₃×C₇	C₂×D₄
prime radical	C₇×#235	C₂
hypercenter	C₂×C₇×Q₈	D₃
hypocenter	C₃	C₇×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×Q₈

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13054722, #13054723, #13054764, …
maximal normal subgroup of	#13054722, #13054723, #13054764, …
maximal quotient of	C₇×#4971, C₇×#4964, C₇×#4966, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 921);