C₂×C₃₇×SD₆₄ #17991078

order	4736
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₃₇×SD₆₄

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₃₇
prime radical	C₂×D₁₆×C₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄×C₃₇ ◁ C₂×C₈×C₃₇ ◁ C₂×C₁₆×C₃₇ ◁ G

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2)(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);(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);(4,18)(5,33)(6,16)(7,31)(8,14)(9,29)(10,12)(11,27)(13,25)(15,23)(17,21)(20,34)(22,32)(24,30)(26,28)

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₃₇×SD₆₄, C₃₇×#13652, C₃₇×#13662, …
maximal normal subgroup of	C₄×C₃₇×SD₆₄, C₃₇×#13652, C₃₇×#13662, …
maximal quotient of	C₂×C₃₇×#1415, C₂×C₃₇×#1416, C₂×C₃₇×#1424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 992);

C2×C37×SD64 #17991078