D₇×C₁₉×C₄₇ #27159758

order	12502
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉×C₄₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₁₉×C₄₇
unambiguous semidirect product	C₇⋊(C₂×C₁₉×C₄₇)

Splittings

Fitting, socle, Hall	C₇×C₁₉×C₄₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₁₉×C₄₇
Hall	C₇×C₄₇ ⋊ C₂×C₁₉
Hall	C₇×C₁₉ ⋊ C₂×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉×C₄₇	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₁₉×C₄₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₉×C₄₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₉×C₄₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6251

A minimal faithful permutation representation — 2 generators on 73 points:

(2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(1,2,4,6,7,5,3)

Conjugacy classes & character table

4465 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×C₁₉×D₄₇, D₇×D₁₉×C₄₇, D₇×D₈₉₃, …
maximal normal subgroup of	D₇×C₁₉×D₄₇, D₇×D₁₉×C₄₇, D₇×D₈₉₃, …
maximal quotient of	C₁₉×Dic₇×C₄₇, C₂×D₇×C₁₉×C₄₇, C₃×D₇×C₁₉×C₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12502, 5);

D7×C19×C47 #27159758