Dic₃₇ #3864

order	148
isoclinism	isoclinic class of D₃₇
automorphism	Aut(G) = C₂×AGL(1,𝔽₃₇) (order 2664) · Inn(G) = D₃₇ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₇

Splittings

derived, Sylow

C₃₇ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₇
commutator, hypocenter	C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃₇	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 148

A minimal faithful permutation representation — 2 generators on 41 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);(1,3,2,4)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^2,37|a^2=b,c^a=c^36

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64438, Dic₇₄, C₄×D₃₇, …
maximal normal subgroup of	#64438, Dic₇₄, C₄×D₃₇, …
maximal quotient of	#64436, C₂×Dic₃₇, C₃×Dic₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(148, 1);

Dic37 #3864