C₂×C₂₅×D₁₃₃ #27210867

order	13300
isoclinism	isoclinic class of D₁₃₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₂₅×D₁₃₃
unambiguous semidirect product	(C₂×C₇)⋊(C₂₅×D₁₉)
unambiguous semidirect product	(C₂×C₁₉)⋊(D₇×C₂₅)

Splittings

Fitting	C₂×C₇×C₁₉×C₂₅ ⋊ C₂
derived, Hall	C₇×C₁₉ ⋊ C₂²×C₂₅
Hall	C₇×C₁₉×C₂₅ ⋊ C₂²
Hall	C₁₉×C₂₅ ⋊ C₂×D₇
Hall	C₇×C₂₅ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×D₇×C₂₅
Sylow	C₇ ⋊ C₂×C₂₅×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₅	D₁₃₃
commutator, hypocenter	C₇×C₁₉	C₂²×C₂₅
Frattini	C₅	C₂×C₅×D₁₃₃
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle	C₂×C₅×C₇×C₁₉	C₂×C₅
prime radical	C₂×C₅×D₁₃₃	C₅
cosocle	C₅×C₇×C₁₉	C₂²×C₅
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂×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)(21,22)(23,24)(25,26)(27,31,39,47,51,29,35,44,50,37,32,40,48,33,42,36,45,30,38,46,41,28,34,43,49);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,23,25,26,24,22)(52,53)

Conjugacy classes & character table

3400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 25);

C2×C25×D133 #27210867