C₇×#19058 #13196274

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#19058

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 840 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) = 896

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

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, 12046);