C₇×#11412 #13188628

order	1792
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇×C₈ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₇×#11412

Subgroups

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

260 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 55
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)(24,25)(26,30)(27,32)(28,33)(29,31)(34,41)(35,42)(36,40)(37,45)(38,43)(39,44)(46,52)(47,50)(48,51)(49,53)(54,55);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)(24,26,27,34,28,35,37,46,29,36,38,47,39,48,49,54)(25,30,32,41,33,42,45,52,31,40,43,50,44,51,53,55);(10,16)(13,19)(17,22)(20,23)(25,31)(30,40)(32,43)(33,44)(41,50)(42,51)(45,53)(52,55)

Conjugacy classes & character table

784 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 4400);