C₃²×D₁₇×C₇⋊D₇ #44472929

order	29988
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₇

Aliases

direct product

C₃²×D₁₇×C₇⋊D₇

Splittings

Fitting, socle, Hall	C₃²×C₇²×C₁₇ ⋊ C₂²
derived, Hall	C₇²×C₁₇ ⋊ C₂²×C₃²
Hall	C₃²×C₇² ⋊ C₂×D₁₇
Hall	C₃²×C₁₇ ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×C₃²×D₁₇
Sylow	C₁₇ ⋊ C₂×C₃²×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₁₇×C₇⋊D₇
commutator, cosocle, hypocenter	C₇²×C₁₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂²
LCS	G ▷ C₇²×C₁₇
UCS	C₁ ◁ C₃²

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 7497

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

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

Conjugacy classes & character table

2340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 570);

C32×D17×C7⋊D7 #44472929