C₇×#18999 #13196215

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#18999

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

850 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 55 points:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 11987);