C₂×Q₈²×C₂₉ #16817191

order	3712
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₂₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×Q₈²×C₂₉

Subgroups

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

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 3712

A minimal faithful permutation representation — 5 generators on 47 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(11,12,13,14)(15,18,17,16);(11,15,13,17)(12,16,14,18)

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₉×#8396, C₂₉×#8451, C₂₉×#12777, …
maximal normal subgroup of	C₂₉×#8396, C₂₉×#8451, C₂₉×#12777, …
maximal quotient of	C₂×Q₈×C₄⋊C₄×C₂₉, C₂×C₂₉×#2369, C₄×Q₈²×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2209);

C2×Q82×C29 #16817191