C₇×#12847 #13190063

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12847

Subgroups

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

434 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 166 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)(10,13)(11,12)(14,15)(16,17,19,24,20,25,30,37,21,23,29,35,31,36,41,45)(18,22,27,33,28,34,40,44,26,32,38,42,39,43,46,47);(9,12)(13,15)(16,18)(17,22)(19,27)(20,28)(21,26)(23,32)(24,33)(25,34)(29,38)(30,40)(31,39)(35,42)(36,43)(37,44)(41,46)(45,47);(8,10,11,14)(9,13,12,15)(17,23)(18,26)(19,29)(25,36)(28,39)(30,41)(33,42)(44,47)

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5835);