D₇×C₁₇×C₃⋊D₂₁ #44472989

order	29988
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	D₇×C₁₇×C₃⋊D₂₁
unambiguous semidirect product	(C₃×D₇)⋊(C₁₇×D₂₁)
unambiguous semidirect product	(C₇×D₇)⋊(C₁₇×C₃⋊D₃)

Splittings

Fitting, socle, Hall	C₃²×C₇²×C₁₇ ⋊ C₂²
derived, Hall	C₃²×C₇² ⋊ C₂²×C₁₇
Hall	C₇²×C₁₇ ⋊ C₂×C₃⋊D₃
Hall	C₃²×C₁₇ ⋊ D₇²
Sylow	C₇² ⋊ C₂×C₁₇×C₃⋊D₃
Sylow	C₃² ⋊ D₇²×C₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₇	D₇×C₃⋊D₂₁
commutator, cosocle, hypocenter	C₃²×C₇²	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂²
LCS	G ▷ C₃²×C₇²
UCS	C₁ ◁ C₁₇

312 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 37
transitive degree	μ_t(G) = 2142

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

2805 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 630);