C₂×#23761611 #34209306

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

Aliases

direct product

C₂×#23761611

Splittings

Sylow

C₇₉ ⋊ C₂×M₁₂₈

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₂×D₇₉
commutator	C₂×C₇₉	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₂²×C₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₃₂
prime radical	C₂³×C₇₉	C₃₂
cosocle	C₃₂×C₇₉	C₂³
hypercenter	C₂²×C₃₂	D₇₉
hypocenter	C₇₉	C₂×M₁₂₈
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 145 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)(83,89)(90,104)(91,105)(92,106)(93,107)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(130,139)(131,140)(132,141)(133,142)(143,145);(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)(82,83,84,90,85,91,94,108,86,92,95,109,98,111,114,130,87,93,96,110,99,112,115,131,101,113,117,132,120,133,134,143,88,89,97,104,100,105,116,124,102,106,118,125,121,127,135,139,103,107,119,126,122,128,136,140,123,129,137,141,138,142,144,145);(80,81)

Conjugacy classes & character table

5152 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90605);

C2×#23761611 #34209306