C₇×C₂₅×Dic₁₉ #27210844

order	13300
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₂₅×Dic₁₉
unambiguous semidirect product	C₁₉⋊(C₄×C₇×C₂₅)

Splittings

derived, Sylow	C₁₉ ⋊ C₄×C₇×C₂₅
Hall	C₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₁₉×C₂₅ ⋊ C₄×C₇
Hall	C₇×C₁₉ ⋊ C₄×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₂₅	D₁₉
commutator, hypocenter	C₁₉	C₄×C₇×C₂₅
Frattini	C₂×C₅	C₅×C₇×D₁₉
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₉	C₂×C₅
cosocle	C₂×C₅×C₁₉	C₂×C₅×C₇
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₇×C₂₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 13300

A minimal faithful permutation representation — 2 generators on 55 points:

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

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, 2);

C7×C25×Dic19 #27210844