C₂×#117×C₃₇ #14557179

order	2368
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#117×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂²
commutator	C₂	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂³
prime radical	C₂⁴×C₃₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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);(3,4,5,6)(7,8)(9,12)(10,11)(13,14);(8,11)(12,14);(7,9,10,13)(8,12,11,14)

Conjugacy classes & character table

1480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1458, C₃₇×#1464, C₃₇×#1729, …
maximal normal subgroup of	C₃₇×#1458, C₃₇×#1464, C₃₇×#1729, …
maximal quotient of	C₂×C₃₇×#376, C₂×C₄×C₂²⋊C₄×C₃₇, C₂×C₄×C₄⋊C₄×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 195);

C2×#117×C37 #14557179