C₁₃²×D₃₇ #27159779

order	12506
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃² · Schur multiplier (H₂) = C₁₃

Aliases

direct product	C₁₃²×D₃₇
unambiguous semidirect product	C₃₇⋊(C₂×C₁₃²)

Splittings

Fitting, socle, Hall	C₁₃²×C₃₇ ⋊ C₂
derived, Sylow	C₃₇ ⋊ C₂×C₁₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃²	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂×C₁₃²
Frattini	C₁	G
Fitting, socle	C₁₃²×C₃₇	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₁₃²

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 6253

A minimal faithful permutation representation — 2 generators on 63 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)(51,52,53,54,55,56,57,58,59,60,61,62,63);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(38,39,40,41,42,43,44,45,46,47,48,49,50)

Conjugacy classes & character table

3380 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₃²×#3866, C₁₃×#14515323, C₁₃×D₁₃×D₃₇, …
maximal normal subgroup of	C₁₃²×#3866, C₁₃×#14515323, C₁₃×D₁₃×D₃₇, …
maximal quotient of	C₁₃²×Dic₃₇, C₂×C₁₃²×D₃₇, C₃×C₁₃²×D₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12506, 10);

C132×D37 #27159779