C₇×#7473 #13184689

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7473

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	D₁₆
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂²×C₇	#357
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

170 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 75 points:

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

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, 461);