C₂×#23761434 #34199442

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

Aliases

direct product

C₂×#23761434

Splittings

Sylow

C₇₉ ⋊ C₂×#1356

Subgroups

	subgroup	G/H
center	C₂²	#19098302
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	#97	C₂²×D₇₉
Fitting	C₂×#347×C₇₉	C₂
socle	C₂²×C₇₉	#329
prime radical	C₂³×C₇₉	#100
cosocle	#97×C₇₉	C₂³
hypercenter	C₂×#347	D₇₉
hypocenter	C₇₉	C₂×#1356
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

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

normal subgroups only

This group has 314 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) = 113
transitive degree	μ_t(G) = 5056

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

Conjugacy classes & character table

1756 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 80741);