C₂×C₇×C₂₅×D₁₉ #27210869

order	13300
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₂₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×C₂₅×D₁₉
unambiguous semidirect product	C₁₉⋊(C₂²×C₇×C₂₅)
unambiguous semidirect product	(C₂×C₁₉)⋊(C₂×C₇×C₂₅)

Splittings

Fitting	C₂×C₇×C₁₉×C₂₅ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂²×C₇×C₂₅
Hall	C₇×C₁₉×C₂₅ ⋊ C₂²
Hall	C₁₉×C₂₅ ⋊ C₂²×C₇
Hall	C₇×C₁₉ ⋊ C₂²×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₂₅	D₁₉
commutator, hypocenter	C₁₉	C₂²×C₇×C₂₅
Frattini	C₅	C₂×C₅×C₇×D₁₉
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle	C₂×C₅×C₇×C₁₉	C₂×C₅
prime radical	C₂×C₅×C₇×D₁₉	C₅
cosocle	C₅×C₁₉	C₂²×C₅×C₇
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₇×C₂₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 6650

A minimal faithful permutation representation — 2 generators on 53 points:

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

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, 27);

C2×C7×C25×D19 #27210869