C₂×C₃₇×#443 #17991771

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×C₃₇×#443

Subgroups

	subgroup	G/H
center	C₂×C₈×C₃₇	D₄
commutator	C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂×#135×C₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 55 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×#1335, C₃₇×#11602, C₂×C₃₇×#1336, …
maximal normal subgroup of	C₂×C₃₇×#1335, C₃₇×#11602, C₂×C₃₇×#1336, …
maximal quotient of	C₂×C₈×D₈×C₃₇, C₂×C₈×SD₁₆×C₃₇, C₂×C₈×Q₁₆×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1685);

C2×C37×#443 #17991771