C₄×#19098286 #34178772

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×#19098286

Splittings

Fitting	C₄×C₂²⋊C₈×C₇₉ ⋊ C₂
Sylow	C₇₉ ⋊ C₄×#323

Subgroups

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂²×C₇₉	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₄×C₂²⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×D₇₉	C₂×C₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₄×C₂²⋊C₈	D₇₉
hypocenter	C₇₉	C₄×#323
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 99 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)(92,94)(93,96)(95,99)(97,98);(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)(84,85,86,88,87,89,90,91)(92,93)(94,98,99,96)(95,97);(80,81,82,83)

Conjugacy classes & character table

3208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 60071);

C4×#19098286 #34178772