C₂×#23761573 #34208626

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

Aliases

direct product

C₂×#23761573

Splittings

Sylow

C₇₉ ⋊ C₂×#1413

Subgroups

	subgroup	G/H
center	C₂²	#19098358
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₇₉
Fitting	C₂×#491×C₇₉	C₂
socle	C₂²×C₇₉	#354
prime radical	C₂³×C₇₉	#99
cosocle	C₄⋊Q₈×C₇₉	C₂³
hypercenter	C₂×#491	D₇₉
hypocenter	C₇₉	C₂×#1413
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ C₂×#491

314 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 83 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:

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

Conjugacy classes & character table

1276 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 89925);