C₃₇×C₂²⋊C₁₆ #14557013

order	2368
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃₇×C₂²⋊C₁₆
unambiguous semidirect product	C₂²⋊(C₁₆×C₃₇)

Subgroups

	subgroup	G/H
center	C₂×C₈×C₃₇	C₂²
commutator	C₂	C₂×C₁₆×C₃₇
Frattini, cosocle	C₂×C₈	C₂²×C₃₇
socle	C₂²×C₃₇	C₂×C₈
prime radical	C₂³×C₃₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₇ ◁ G

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39)(40,41)(42,43,44,47,45,48,50,53,46,49,51,54,52,55,56,57);(39,41)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#1314, C₃₇×#1315, C₃₇×#1322, …
maximal normal subgroup of	C₃₇×#1314, C₃₇×#1315, C₃₇×#1322, …
maximal quotient of	C₃₇×#1314, C₃₇×#1322, C₃₇×D₄⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 29);

C37×C22⋊C16 #14557013