C₇×#12840 #13190056

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12840

Subgroups

	subgroup	G/H
center	C₄×C₇	#427
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1808
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ C₂×C₇×#131 ◁ G

354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 3 generators on 71 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5828);