C₂×#125×C₃₇ #14557192

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

Aliases

direct product

C₂×#125×C₃₇

Subgroups

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

226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 2368

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#1470, C₂×C₃₇×#330, C₃₇×#1528, …
maximal normal subgroup of	C₃₇×#1470, C₂×C₃₇×#330, C₃₇×#1528, …
maximal quotient of	C₂×C₃₇×#382, C₂×C₃₇×#387, C₂×C₃₇×#389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 208);

C2×#125×C37 #14557192