C₂×C₅²×C₇×D₁₉ #27210932

order	13300
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅²×C₇ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₂×C₅²×C₇×D₁₉
unambiguous semidirect product	C₁₉⋊(C₂²×C₅²×C₇)
unambiguous semidirect product	(C₂×C₁₉)⋊(C₂×C₅²×C₇)

Splittings

Fitting, socle	C₂×C₅²×C₇×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂²×C₅²×C₇
Hall	C₅²×C₇×C₁₉ ⋊ C₂²
Hall	C₅²×C₁₉ ⋊ C₂²×C₇
Hall	C₇×C₁₉ ⋊ C₂²×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₇	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂²×C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅²×C₇

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 6650

A minimal faithful permutation representation — 2 generators on 38 points:

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

Conjugacy classes & character table

3850 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 90);

C2×C52×C7×D19 #27210932