#1358×D₇₉ #34199474

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

Aliases

direct product

#1358×D₇₉

Splittings

Fitting	C₇₉×#1358 ⋊ C₂
Sylow	C₇₉ ⋊ C₂×#1358

Subgroups

	subgroup	G/H
center	C₂	#331×D₇₉
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1358	C₂
socle	C₂×C₇₉	C₂×#331
prime radical	C₂²×D₇₉	#100
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1358	D₇₉
hypocenter	C₇₉	C₂×#1358
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1358

390 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 111
transitive degree	μ_t(G) = 2528

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 80773);