C₂²×C₂₉×Q₃₂ #16817124

order	3712
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×C₂₉×Q₃₂

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₂₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×C₈×C₂₉ ◁ G

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 4 generators on 65 points:

(1,2)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22);(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#13015, C₂₉×#13020, C₂₉×#13236, …
maximal normal subgroup of	C₂₉×#13015, C₂₉×#13020, C₂₉×#13236, …
maximal quotient of	C₂²×C₂₉×#358, C₂²×C₂₉×#366, C₂×C₄×C₂₉×Q₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2142);

C22×C29×Q32 #16817124