C₇×#13633 #13190849

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#13633

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	C₂³×C₇	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 190 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)(15,21)(16,34)(17,38)(18,40)(19,24)(20,28)(22,26)(23,35)(27,47)(29,60)(30,62)(31,50)(32,63)(33,46)(36,66)(37,65)(39,67)(41,55)(42,69)(43,59)(44,52)(45,54)(48,51)(49,57)(53,64)(56,72)(58,68)(61,75)(70,73)(71,74);(12,14,26,51,17,29,32,55,15,27,52,73,33,56,37,59,19,31,54,74,39,61,36,58,18,30,53,69,35,57,16,28)(13,20,34,49,23,42,64,62,40,68,66,75,67,71,45,50,24,43,65,72,46,70,44,47,21,41,63,60,38,48,22,25);(12,15,19,18)(13,21,24,40)(14,27,31,30)(16,32,37,36)(17,33,39,35)(20,41,43,68)(22,44,45,64)(23,38,46,67)(25,47,50,62)(26,52,54,53)(28,55,59,58)(29,56,61,57)(34,63,65,66)(42,48,70,71)(49,60,72,75)(51,73,74,69)

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, 6621);