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C₂×C₁₇×(C₇⋊C₉)⋊D₇ #44472573

order	29988
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₁₇×(C₇⋊C₉)⋊D₇
unambiguous semidirect product	(C₂×C₇⋊C₉)⋊(D₇×C₁₇)

Splittings

derived, Sylow	C₇² ⋊ C₂²×C₉×C₁₇
Hall	C₇×C₁₇×C₇⋊C₉ ⋊ C₂²
Hall	C₂×C₁₇×C₇⋊D₇ ⋊ C₉
Hall	C₇²×C₁₇ ⋊ C₂²×C₉
Hall	C₇×C₇⋊C₉ ⋊ C₂²×C₁₇
Hall	C₂×C₇⋊D₇ ⋊ C₉×C₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₇	ΓL(1,𝔽₈)⋊D₇
commutator, hypocenter	C₇²	C₂²×C₉×C₁₇
Frattini	C₃	C₂×C₁₇×ΓL(1,𝔽₈)⋊D₇
Fitting, socle	C₂×C₃×C₇²×C₁₇	C₂×C₃
prime radical	C₂×C₃×C₁₇×C₇⋊D₇	C₃
cosocle	C₃×C₇²	C₂²×C₃×C₁₇
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₃×C₁₇

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

normal subgroups only

This group has 160 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

transitive degree

μ_t(G) = 2142

Conjugacy classes & character table

2244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 214);