C₂×#20261637 #27157984

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20261637

Splittings

Fitting	C₂³×C₄×C₉₇ ⋊ C₄
Sylow	C₉₇ ⋊ C₂×#352

Subgroups

	subgroup	G/H
center	C₂²	#15690868
commutator	C₂³×C₉₇	C₂²×C₄
Frattini	C₂³	C₂²×#87604
Fitting	C₂³×C₄×C₉₇	C₄
socle	C₂²×C₉₇	#99
prime radical	C₂²×C₂²⋊D₉₇	C₂²
cosocle	C₂×C₂²⋊D₉₇	C₂³
hypercenter	C₂³×C₄	#87604
hypocenter	C₉₇	C₂×#352
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄

278 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 3104

A minimal faithful permutation representation — 3 generators on 115 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20758);