C₇×#13618 #13190834

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13618

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₂³×C₇	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

290 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 75
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 75 points:

(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,25)(15,20)(16,34)(17,38)(18,24)(19,27)(21,36)(22,32)(23,33)(26,53)(28,59)(29,55)(30,54)(31,64)(35,50)(37,52)(39,47)(40,46)(41,56)(42,69)(43,62)(44,58)(45,66)(48,75)(49,57)(51,60)(61,73)(63,74)(65,72)(67,68)(70,71);(12,14,18,29)(13,19,24,44)(15,30,33,26)(16,35,37,70)(17,39,40,63)(20,43,23,41)(21,47,49,74)(22,50,51,71)(25,52,55,34)(27,57,58,36)(28,60,61,32)(31,65,67,69)(38,42,46,72)(45,59,48,73)(53,64,54,68)(56,66,62,75);(12,15,18,33)(13,20,24,23)(14,26,29,30)(16,31,37,67)(17,32,40,60)(19,41,44,43)(21,45,49,48)(22,46,51,38)(25,53,55,54)(27,56,58,62)(28,39,61,63)(34,64,52,68)(35,69,70,65)(36,66,57,75)(42,71,72,50)(47,73,74,59)

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, 6606);