C₄×#19098398 #34208841

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₄×#19098398

Splittings

Fitting	C₄×D₁₆×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₄×D₃₂

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579102
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×D₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂×#19098398	C₂
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₄×D₁₆	D₇₉
hypocenter	C₇₉	C₄×D₃₂
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×D₁₆

214 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 10112

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

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

Conjugacy classes & character table

1792 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90140);