C₇×#12465 #13189681

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12465

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

196 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(9,15)(10,20)(11,16)(12,27)(13,26)(18,38)(19,28)(21,22)(23,25)(24,32)(29,34)(30,31)(33,39)(35,36);(8,9,16,31,13,18,12,28,14,19,27,38,26,30,11,15)(10,17,32,34,22,35,21,33,24,37,20,39,23,36,25,29);(8,10)(9,17)(11,25)(12,21)(13,22)(14,24)(15,29)(16,32)(18,35)(19,37)(20,27)(23,26)(28,33)(30,36)(31,34)(38,39)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5453);