C₂×#23761326 #34197338

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#23761326

Splittings

Sylow

C₇₉ ⋊ C₂×#1315

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#408×C₇₉	C₂
socle	C₂³×C₇₉	C₂²⋊C₈
prime radical	C₂²×#41×C₇₉	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#408	D₇₉
hypocenter	C₇₉	C₂×#1315
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#408

534 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 121
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 121 points:

(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)(82,84)(83,86)(85,89)(87,88)(90,92)(91,97)(93,106)(94,108)(95,103)(96,105)(98,114)(99,116)(100,102)(101,113)(104,110)(107,120)(109,111)(112,118)(115,119)(117,121);(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)(82,83)(84,88,89,86)(85,87)(90,91,94,99,95,100,109,117,96,101,110,118,93,98,107,115)(92,102,104,119,103,113,120,116,105,114,108,121,106,97,111,112);(80,81)

Conjugacy classes & character table

2260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78637);