C₇×#12031 #13189247

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇×#12031

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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)(26,35,38,50)(27,32)(28,41)(29,44)(30,34)(31,37,46,53)(33,42)(36,49,39,51)(40,48)(43,55)(45,52,47,54);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)(24,26,28,37,29,36,42,52,30,38,43,53,27,39,40,54)(25,31,33,35,32,47,41,51,34,46,48,50,44,45,55,49);(10,16)(13,19)(17,22)(20,23)(24,27,30,29)(25,32,34,44)(26,36,38,39)(28,40,43,42)(31,45,46,47)(33,41,48,55)(35,49,50,51)(37,52,53,54)

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5019);