C₂×C₇×#1415 #13190829

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₇×#1415

Subgroups

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

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

normal subgroups only

This group has 342 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) = 896

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

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

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, 6601);