C₇×D₈₉₃ #27159757

order	12502
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₈₉₃
unambiguous semidirect product	C₁₉⋊(C₇×D₄₇)
unambiguous semidirect product	C₄₇⋊(C₇×D₁₉)

Splittings

Fitting, socle, Hall	C₇×C₁₉×C₄₇ ⋊ C₂
derived, Hall	C₁₉×C₄₇ ⋊ C₂×C₇
Hall	C₇×C₄₇ ⋊ D₁₉
Hall	C₇×C₁₉ ⋊ D₄₇
Sylow	C₄₇ ⋊ C₇×D₁₉
Sylow	C₁₉ ⋊ C₇×D₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	D₈₉₃
commutator, cosocle, hypocenter	C₁₉×C₄₇	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₉×C₄₇	C₂
LCS	G ▷ C₁₉×C₄₇
UCS	C₁ ◁ C₇

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

Conjugacy classes & character table

3136 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×D₁₉×D₄₇, D₇×D₈₉₃, D₁₃₃⋊D₄₇, …
maximal normal subgroup of	C₇×D₁₉×D₄₇, D₇×D₈₉₃, D₁₃₃⋊D₄₇, …
maximal quotient of	C₇×Dic₈₉₃, C₂×C₇×D₈₉₃, C₃×C₇×D₈₉₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12502, 4);

C7×D893 #27159757