C₂²×C₄×Q₈×C₂₉ #16817137

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×C₄×Q₈×C₂₉

Subgroups

	subgroup	G/H
center	C₂³×C₄×C₂₉	C₂²
commutator	C₂	C₂⁴×C₄×C₂₉
Frattini, cosocle	C₂²	C₂⁵×C₂₉
socle, prime radical	C₂⁴×C₂₉	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₉ ◁ G

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 3712

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);(5,6,7,8);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8303, C₂₉×#8311, C₂₉×#10763, …
maximal normal subgroup of	C₂₉×#8303, C₂₉×#8311, C₂₉×#10763, …
maximal quotient of	C₂²×C₄×C₄⋊C₄×C₂₉, C₂×C₄²×Q₈×C₂₉, C₂²×C₂₉×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2155);

C22×C4×Q8×C29 #16817137