C₂×C₇×#1416 #13190830

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₇×#1416

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

302 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6602);