C₇×#13615 #13190831

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13615

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂×D₁₆
prime radical	C₂×C₇×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 298 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)(26,52)(28,58)(29,60)(30,53)(31,61)(35,49)(37,67)(39,46)(40,45)(41,54)(42,68)(43,57)(44,63)(47,73)(48,70)(50,65)(51,56)(55,72)(59,74)(62,71)(64,66)(69,75);(12,14)(13,19)(15,26)(16,35)(17,39)(18,30)(20,41)(21,46)(22,49)(23,51)(24,43)(25,34)(27,36)(28,32)(29,33)(31,62)(37,69)(38,42)(40,55)(44,58)(45,71)(47,74)(48,72)(50,75)(52,61)(53,67)(54,63)(56,73)(57,70)(59,65)(60,66)(64,68);(12,15,18,33)(13,20,24,23)(14,26,30,29)(16,31,37,64)(17,32,40,65)(19,41,43,51)(21,44,48,47)(22,45,50,38)(25,52,53,60)(27,54,57,56)(28,55,59,39)(34,61,67,66)(35,62,69,68)(36,63,70,73)(42,49,71,75)(46,58,72,74)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6603);