C₂²×C₂₉×#117 #16817135

order	3712
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₂₉ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×C₂₉×#117

Subgroups

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

1880 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);(5,6,7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(9,11,12,15)(10,14,13,16)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#8300, C₂₉×#10713, C₂₉×#10716, …
maximal normal subgroup of	C₂₉×#8300, C₂₉×#10713, C₂₉×#10716, …
maximal quotient of	C₂²×C₂₉×#376, C₂²×C₄×C₂²⋊C₄×C₂₉, C₂²×C₄×C₄⋊C₄×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 2153);

C22×C29×#117 #16817135