C₄×#126×C₉₇ #27138262

order	12416
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₉₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#126×C₉₇

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₉₇	C₂³
commutator	C₂²	C₂³×C₄×C₉₇
Frattini, cosocle	C₂³	C₂⁴×C₉₇
socle	C₂³×C₉₇	C₂⁴
prime radical	C₂⁴×C₉₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉₇ ◁ G

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 117
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 4 generators on 117 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97)(98,99,100,101);(102,103)(104,106)(105,107)(108,109)(110,111)(112,116)(113,114)(115,117);(102,104,105,108)(103,106,107,109)(111,114)(115,116);(103,107)(106,109)(110,112,113,117)(111,115,114,116)

Conjugacy classes & character table

5432 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 1036);