C₂₅×Dic₁₃₃ #27210846

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₅×Dic₁₃₃
unambiguous semidirect product	C₇⋊(C₂₅×Dic₁₉)
unambiguous semidirect product	C₁₉⋊(C₂₅×Dic₇)

Splittings

derived, Hall	C₇×C₁₉ ⋊ C₄×C₂₅
Hall	C₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₁₉×C₂₅ ⋊ Dic₇
Hall	C₇×C₂₅ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ C₂₅×Dic₇
Sylow	C₇ ⋊ C₂₅×Dic₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₅	D₁₃₃
commutator, hypocenter	C₇×C₁₉	C₄×C₂₅
Frattini	C₂×C₅	C₅×D₁₃₃
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle, prime radical, cosocle	C₂×C₅×C₇×C₁₉	C₂×C₅
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂×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)(21,22)(23,24)(25,26)(27,28,29,30)(31,32,34,37,41,33,35,38,42,46,36,39,43,47,50,40,44,48,51,53,45,49,52,54,55);(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)

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, 4);

C25×Dic133 #27210846