#1356×D₇₉ #34199441

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

Aliases

direct product

#1356×D₇₉

Splittings

Fitting	C₇₉×#1356 ⋊ C₂
Sylow	C₇₉ ⋊ C₂×#1356

Subgroups

	subgroup	G/H
center	C₂	#329×D₇₉
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	#97	C₂²×D₇₉
Fitting	C₇₉×#1356	C₂
socle	C₂×C₇₉	C₂×#329
prime radical	C₂²×D₇₉	#100
cosocle	#97×C₇₉	C₂³
hypercenter	#1356	D₇₉
hypocenter	C₇₉	C₂×#1356
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #1356

314 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 71 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)(81,86,90,103)(83,95,96,100)(84,98,99,93)(87,102,89,91)(88,106,108,109)(92,97,110,104)(101,105,111,107);(80,81,83,88,84,89,97,107,85,90,96,108,99,87,104,105)(82,91,92,109,93,86,100,101,94,102,110,106,98,103,95,111)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 80740);