C₂×#23761338 #34197394

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

Aliases

direct product

C₂×#23761338

Splittings

Sylow

C₇₉ ⋊ C₂×#1321

Subgroups

	subgroup	G/H
center	C₂×C₄	#14579076
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×M₁₆	C₂²×D₇₉
Fitting	C₂×#408×C₇₉	C₂
socle	C₂²×C₇₉	#323
prime radical	C₂²×#41×C₇₉	C₄
cosocle	C₂×M₁₆×C₇₉	C₂³
hypercenter	C₂×#408	D₇₉
hypocenter	C₇₉	C₂×#1321
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#408

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 113 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)(83,89)(85,98)(86,100)(87,94)(88,96)(90,106)(91,104)(92,105)(93,97)(95,102)(99,103)(101,111)(107,112)(108,110)(109,113);(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)(82,83,86,91,87,90,101,107,88,92,102,109,85,97,99,108)(84,93,95,110,94,89,103,104,96,106,100,112,98,105,111,113);(80,81)

Conjugacy classes & character table

2236 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 78693);

C2×#23761338 #34197394