C₂×2₋¹⁺⁴×C₃₇ #14557249

order	2368
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×2₋¹⁺⁴×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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);(4,8)(6,14)(9,11)(12,18);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)

Conjugacy classes & character table

1258 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1793, C₃₇×#1796, C₃₇×#2884, …
maximal normal subgroup of	C₃₇×#1793, C₃₇×#1796, C₃₇×#2884, …
maximal quotient of	C₂×C₃₇×#519, C₂×C₃₇×#520, C₄×2₋¹⁺⁴×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 265);

C2×2−1+4×C37 #14557249