C₇×#8142 #13185358

order	1792
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8142

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

294 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 448

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

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

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, 1130);