D₄×M₁₆×C₃₇ #17991752

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×M₁₆×C₃₇

Subgroups

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

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 49 points:

(1,2,3,4)(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);(2,4);(5,6,7,8,9,10,11,12);(6,10)(8,12)

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8906, C₃₇×#8908, C₃₇×#8989, …
maximal normal subgroup of	C₃₇×#8906, C₃₇×#8908, C₃₇×#8989, …
maximal quotient of	M₁₆×D₈×C₃₇, C₃₇×#18035, M₁₆×SD₁₆×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1666);

D4×M16×C37 #17991752