C₇×#7449 #13184665

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7449

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1361
commutator	C₂×C₁₆	C₂×C₄×C₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
prime radical	C₂²×C₇	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

126 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,18,35,21,37,47,15,22,36,20,14,38,13,19)(10,23,25,54,39,56,63,71,28,57,62,46,27,64,26,55)(11,29,31,17,45,48,58,50,34,66,69,51,33,49,32,65)(16,30,40,24,43,59,52,70,44,68,42,61,53,67,41,60);(8,10,15,28)(9,16,22,44)(11,30,34,68)(12,26,36,63)(13,25,37,62)(14,39,35,27)(17,46,51,54)(18,41,20,52)(19,40,47,42)(21,53,38,43)(23,48,57,49)(24,58,61,32)(29,64,66,56)(31,60,69,70)(33,59,45,67)(50,71,65,55)

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 437);