C₂²⋊C₃₂×D₇₉ #34208458

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂²⋊C₃₂×D₇₉

Splittings

Fitting	C₇₉×C₂²⋊C₃₂ ⋊ C₂
Sylow	C₇₉ ⋊ C₂×C₂²⋊C₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₉
commutator	C₂×C₇₉	C₂²×C₃₂
Fitting	C₇₉×C₂²⋊C₃₂	C₂
socle	C₂²×C₇₉	C₂²×C₁₆
prime radical	C₂³×D₇₉	C₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	C₂²⋊C₃₂	D₇₉
hypocenter	C₇₉	C₂×C₂²⋊C₃₂
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²⋊C₃₂

342 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 115 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)(81,83);(80,81)(82,83)(84,85,86,90,87,91,94,100,88,92,95,101,97,103,106,110,89,93,96,102,98,104,107,111,99,105,108,112,109,113,114,115)

Conjugacy classes & character table

3280 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 89757);