C₁₉×Dic₁₇₅ #27210849

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×Dic₁₇₅
unambiguous semidirect product	C₇⋊(C₁₉×Dic₂₅)

Splittings

derived, Hall	C₇×C₂₅ ⋊ C₄×C₁₉
Hall	C₇×C₁₉×C₂₅ ⋊ C₄
Hall	C₁₉×C₂₅ ⋊ Dic₇
Hall	C₇×C₁₉ ⋊ Dic₂₅
Sylow	C₂₅ ⋊ C₁₉×Dic₇
Sylow	C₇ ⋊ C₁₉×Dic₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	D₁₇₅
commutator, hypocenter	C₇×C₂₅	C₄×C₁₉
Frattini	C₂×C₅	C₁₉×D₃₅
Fitting	C₂×C₇×C₁₉×C₂₅	C₂
socle, prime radical	C₂×C₅×C₇×C₁₉	D₅
cosocle	C₂×C₇×C₂₅	C₂×C₁₉
LCS	G ▷ C₇×C₂₅
UCS	C₁ ◁ C₂×C₁₉

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

Conjugacy classes & character table

3382 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 7);

C19×Dic175 #27210849