C₂²×C₂₉×C₂²⋊Q₈ #16817147

order	3712
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₂₉ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×C₂₉×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₂₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₉
prime radical	C₂⁵×C₂₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₉ ◁ G

1840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1856

A minimal faithful permutation representation — 5 generators on 45 points:

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

Conjugacy classes & character table

1624 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8308, C₂₉×#8329, C₂₉×#8550, …
maximal normal subgroup of	C₂₉×#8308, C₂₉×#8329, C₂₉×#8550, …
maximal quotient of	C₂²×C₂₉×#380, C₂²×C₂₉×#385, C₂²×C₂₉×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2165);

C22×C29×C22⋊Q8 #16817147