D₁₇×(C₇⋊C₉)⋊D₇ #44472546

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

Aliases

direct product	D₁₇×(C₇⋊C₉)⋊D₇
unambiguous semidirect product	(D₁₇×C₇⋊C₉)⋊D₇

Splittings

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

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₁₇×ΓL(1,𝔽₈)⋊D₇
commutator, hypocenter	C₇²×C₁₇	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₇	C₂²×C₃
prime radical	C₃×D₁₇×C₇⋊D₇	C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁ ◁ C₃

160 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 40
transitive degree	μ_t(G) = 1071

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 187);