C₇×#12889 #13190105

order	1792
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#12889

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 51
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,34)(23,28)(24,42)(25,45)(26,46)(27,33)(29,38)(30,35)(31,44)(32,43)(36,48)(37,47)(39,51)(40,50)(41,49);(13,16)(17,19)(20,22,35,40,26,37,44,41,27,38,25,36,43,39,24,23)(21,28,42,51,32,48,45,29,33,49,31,47,46,50,30,34);(12,14,15,18)(13,17,16,19)(20,23,24,39,43,36,25,38,27,41,44,37,26,40,35,22)(21,29,30,48,46,51,31,28,33,34,45,50,32,47,42,49)

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, 5877);