C₇₁×#7383 #31828255

order	18176
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7383

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₁	D₁₆
commutator	C₁₆	C₂×C₈×C₇₁
Frattini, cosocle	C₄×C₁₆	C₂²×C₇₁
socle	C₂²×C₇₁	#357
prime radical	C₂×D₁₆×C₇₁	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₄²×C₇₁ ◁ C₄×C₈×C₇₁ ◁ C₄×C₁₆×C₇₁ ◁ G

138 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 111
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 2 generators on 111 points:

(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,24)(13,27)(14,19)(16,20)(17,21)(18,25)(22,34)(23,30)(26,40)(28,36)(29,35)(31,39)(32,37)(33,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(10,15)(11,20)(12,21)(13,25)(16,17)(18,34)(19,30)(22,39)(23,37)(24,28)(26,38)(27,29)(31,40)(32,33)(35,36)

Conjugacy classes & character table

5396 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 371);