C₇₁×#7186 #31828058

order	18176
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7186

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₇₁
Frattini, cosocle	C₂³×C₈	C₂²×C₇₁
socle	C₂³×C₇₁	C₂²⋊C₈
prime radical	C₂⁴×C₇₁	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ C₂³×C₈×C₇₁ ◁ G

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

normal subgroups only

This group has 326 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) = 115
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 2 generators on 115 points:

(5,6)(7,11,12,9)(8,10)(13,14,17,22,18,23,32,40,19,24,33,41,16,21,30,38)(15,25,27,42,26,36,43,39,28,37,31,44,29,20,34,35)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,29)(17,31)(18,26)(19,28)(21,37)(22,39)(23,25)(24,36)(27,33)(30,43)(32,34)(35,41)(38,42)(40,44)

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, 174);