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C₇×#3675 #11758917

order	1008
isoclinism	isoclinic class of #3672
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₇×Q₈×C₉ ⋊ C₂
Hall	C₇×C₉ ⋊ SD₁₆
Sylow	C₉ ⋊ C₇×SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₇	#611
commutator, cosocle	C₄×C₉	C₂²×C₇
Frattini	C₃×C₄	C₂×D₃×C₇
Fitting	C₇×Q₈×C₉	C₂
socle	C₂×C₃×C₇	#67
prime radical	C₇×D₃₆	C₂
hypercenter	C₇×Q₈	D₉
hypocenter	C₉	C₇×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×Q₈

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 88);