C₂²×C₂₉×M₃₂ #16817119

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×C₂₉×M₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₂₉	C₂²
commutator	C₂	C₂³×C₈×C₂₉
Frattini, cosocle	C₈	C₂⁴×C₂₉
socle	C₂³×C₂₉	C₂×C₈
prime radical	C₂⁴×C₂₉	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₉ ◁ G

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 4 generators on 49 points:

(1,2)(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);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,14)(8,16)(10,18)(12,20)

Conjugacy classes & character table

2320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#12544, C₂₉×#12545, C₂×C₂₉×#1375, …
maximal normal subgroup of	C₂₉×#12544, C₂₉×#12545, C₂×C₂₉×#1375, …
maximal quotient of	C₂²×C₂₉×#346, C₂×C₄×C₂₉×M₃₂, C₂²×C₂₉×C₂²⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2137);

C22×C29×M32 #16817119