C₇₁×#7135 #31828007

order	18176
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₇₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₁×#7135

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#333
commutator	C₄²	C₂×C₈×C₇₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₁
prime radical	C₂³×C₇₁	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

202 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 4544

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

(1,2)(3,7,8,5)(4,6)(9,10,12,17,13,16,25,33,14,19,26,36,28,18,37,35)(11,20,22,34,23,15,29,30,21,31,38,40,27,32,24,39)(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);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15)(12,24,26,29)(13,27,28,23)(16,20)(17,34)(18,31)(19,32)(22,25,38,37)(30,35)(33,39)(36,40)

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 123);