C₇×#11428 #13188644

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#11428

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#99
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

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

normal subgroups only

This group has 526 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4416);