C₂×#124×C₃₇ #14557191

order	2368
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#124×C₃₇

Subgroups

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

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(4,5)(9,13)(11,14);(3,4,6,5)(7,8,10,12)(9,11,13,14);(3,5,6,4)(7,9)(8,11)(10,13)(12,14)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1469, C₂×C₃₇×#329, C₃₇×#1527, …
maximal normal subgroup of	C₃₇×#1469, C₂×C₃₇×#329, C₃₇×#1527, …
maximal quotient of	C₂×C₃₇×#382, C₂×C₃₇×#388, C₂×C₃₇×#390, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 207);

C2×#124×C37 #14557191