C₂×C₄×Q₁₆×C₃₇ #17991756

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×Q₁₆×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	D₄
commutator	C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle, prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 4 generators on 59 points:

(1,2)(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);(3,4,5,6);(7,8,9,10,11,12,13,14)(15,22,21,20,19,18,17,16);(7,15,11,19)(8,16,12,20)(9,17,13,21)(10,18,14,22)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8946, C₃₇×#8950, C₃₇×#9656, …
maximal normal subgroup of	C₃₇×#8946, C₃₇×#8950, C₃₇×#9656, …
maximal quotient of	C₂×C₈×Q₁₆×C₃₇, C₂×C₃₇×#1584, C₂×C₃₇×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1670);

C2×C4×Q16×C37 #17991756