C₇×#23357 #13200573

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#23357

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂×C₇×#127	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

912 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 306 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) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,14)(11,15)(12,13)(16,17)(18,22)(19,21)(20,23)(24,25,28,31,27,29,26,30);(9,13)(10,11)(18,23)(20,22)(25,29)(30,31);(8,10,12,11)(9,14,13,15)(16,18,19,23)(17,20,21,22)(24,26,27,28)(25,30,29,31);(8,11,12,10)(9,14,13,15)(17,21)(20,22)(24,26,27,28)(25,30,29,31)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 16345);