C₂×C₁₇⋊((C₇⋊C₉)⋊D₇) #44472574

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

Aliases

direct product	C₂×C₁₇⋊((C₇⋊C₉)⋊D₇)
unambiguous semidirect product	(C₂×C₁₇)⋊((C₇⋊C₉)⋊D₇)
unambiguous semidirect product	(C₂×C₇⋊C₉)⋊D₁₁₉

Splittings

derived, Hall	C₇²×C₁₇ ⋊ C₂²×C₉
Hall	C₇×C₁₇×C₇⋊C₉ ⋊ C₂²
Hall	C₂×C₇⋊D₁₁₉ ⋊ C₉
Hall	C₇×C₇⋊C₉ ⋊ C₂×D₁₇
Sylow	C₇² ⋊ C₂×C₉×D₁₇
Sylow	C₁₇ ⋊ C₂×(C₇⋊C₉)⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	C₁₇⋊(ΓL(1,𝔽₈)⋊D₇)
commutator, hypocenter	C₇²×C₁₇	C₂²×C₉
Frattini	C₃	C₂×C₁₇⋊(ΓL(1,𝔽₈)⋊D₇)
Fitting, socle	C₂×C₃×C₇²×C₁₇	C₂×C₃
prime radical	C₂×C₃×C₇⋊D₁₁₉	C₃
cosocle	C₃×C₇²×C₁₇	C₂²×C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁ ◁ C₂×C₃

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 42
transitive degree	μ_t(G) = 2142

A minimal faithful permutation representation — 2 generators on 42 points:

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

Conjugacy classes & character table

1812 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 215);