C₇×C₁₉⋊#1043 #27210895

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₁₉⋊#1043
unambiguous semidirect product	C₁₉⋊(C₇×#1043)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₁₉ ⋊ C₄
derived, Hall	C₅²×C₁₉ ⋊ C₄×C₇
Hall	C₅²×C₇ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ #1043
Sylow	C₅² ⋊ C₇×Dic₁₉
Sylow	C₁₉ ⋊ C₇×#1043

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₁₉⋊#1043
commutator, hypocenter	C₅²×C₁₉	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₁₉	C₄
prime radical	C₇×C₁₉×C₅⋊D₅	C₂
cosocle	C₁₉×C₅⋊D₅	C₂×C₇
LCS	G ▷ C₅²×C₁₉
UCS	C₁ ◁ C₇

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 3325

A minimal faithful permutation representation — 3 generators on 36 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 53);

C7×C19⋊#1043 #27210895