C₂²×C₇₉⋊C₆₄ #34209303

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₆₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂²×C₇₉⋊C₆₄

Splittings

derived, Sylow

C₇₉ ⋊ C₂²×C₆₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃₂	D₇₉
commutator, hypocenter	C₇₉	C₂²×C₆₄
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₃₂
cosocle	C₃₂×C₇₉	C₂³
LCS	G ▷ C₇₉
UCS	C₁ ◁ C₂²×C₃₂

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 20224

A minimal faithful permutation representation — 3 generators on 147 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)(84,85,86,91,87,92,96,106,88,93,97,107,100,110,116,126,89,94,98,108,101,111,117,127,103,113,119,129,122,132,136,141,90,95,99,109,102,112,118,128,104,114,120,130,123,133,137,142,105,115,121,131,124,134,138,143,125,135,139,144,140,145,146,147);(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);(80,81)

Conjugacy classes & character table

5248 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90602);

C22×C79⋊C64 #34209303