C₂×#23761612 #34209311

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

Aliases

direct product

C₂×#23761612

Splittings

Fitting	C₂×D₃₂×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₂×D₆₄

Subgroups

	subgroup	G/H
center	C₂²	#19098398
commutator, cosocle	C₃₂×C₇₉	C₂³
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₂×D₃₂×C₇₉	C₂
socle	C₂²×C₇₉	D₃₂
hypercenter	C₂×D₃₂	D₇₉
hypocenter	C₇₉	C₂×D₆₄
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×D₃₂

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 145
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 145 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)(83,89)(84,95)(85,96)(86,101)(87,105)(90,91)(92,117)(93,119)(94,111)(97,127)(98,130)(99,123)(100,108)(102,136)(103,126)(104,109)(106,134)(107,110)(112,132)(113,131)(114,116)(115,138)(118,142)(120,121)(122,128)(124,125)(129,145)(133,140)(135,139)(137,141)(143,144);(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)(82,83)(84,89)(85,100)(86,104)(87,107)(88,94)(90,95)(91,96)(92,101)(93,105)(97,117)(98,119)(99,111)(102,135)(103,133)(106,137)(108,121)(109,120)(110,128)(112,127)(113,130)(114,123)(115,136)(116,126)(118,134)(122,138)(124,132)(125,131)(129,142)(139,145)(140,144)(141,143);(80,81)

Conjugacy classes & character table

1552 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90610);