D₅×C₁₃×C₉₇ #27162170

order	12610
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃×C₉₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₃×C₉₇
unambiguous semidirect product	C₅⋊(C₂×C₁₃×C₉₇)

Splittings

Fitting, socle, Hall	C₅×C₁₃×C₉₇ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₃×C₉₇
Hall	C₅×C₉₇ ⋊ C₂×C₁₃
Hall	C₅×C₁₃ ⋊ C₂×C₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃×C₉₇	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₃×C₉₇
Frattini	C₁	G
Fitting, socle	C₅×C₁₃×C₉₇	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₃×C₉₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 6305

A minimal faithful permutation representation — 2 generators on 115 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)(103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,4,5,3)

Conjugacy classes & character table

5044 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃×AGL(1,𝔽₅)×C₉₇, C₁₃×#14515596, C₉₇×#63120, …
maximal normal subgroup of	C₁₃×AGL(1,𝔽₅)×C₉₇, C₁₃×#14515596, C₉₇×#63120, …
maximal quotient of	C₁₃×Dic₅×C₉₇, C₂×D₅×C₁₃×C₉₇, C₃×D₅×C₁₃×C₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12610, 5);

D5×C13×C97 #27162170