Q₁₂₈×D₇₉ #34209317

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

Aliases

direct product

Q₁₂₈×D₇₉

Splittings

Fitting	C₇₉×Q₁₂₈ ⋊ C₂
Sylow	C₇₉ ⋊ C₂×Q₁₂₈

Subgroups

	subgroup	G/H
center	C₂	D₃₂×D₇₉
commutator, cosocle	C₃₂×C₇₉	C₂³
Frattini	C₃₂	C₂²×D₇₉
Fitting	C₇₉×Q₁₂₈	C₂
socle	C₂×C₇₉	C₂×D₃₂
prime radical	C₂×D₇₉	D₃₂
hypercenter	Q₁₂₈	D₇₉
hypocenter	C₇₉	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 207 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)(80,82,87,99)(81,88,93,122)(83,101,106,154)(84,108,111,171)(85,113,115,175)(86,117,114,98)(89,124,127,188)(90,129,131,196)(91,133,134,198)(92,135,116,121)(94,136,139,100)(95,123,118,103)(96,128,144,102)(97,132,146,109)(104,161,163,207)(105,165,162,153)(107,119,168,181)(110,173,156,170)(112,120,174,183)(125,190,191,204)(126,192,176,187)(130,197,166,195)(137,201,202,160)(138,150,147,164)(140,189,178,206)(141,159,177,186)(142,199,203,148)(143,157,169,193)(145,200,205,149)(151,185,182,155)(152,184,179,158)(167,194,172,180);(80,81,87,93)(82,94,99,139)(83,100,106,136)(84,107,111,168)(85,112,115,174)(86,116,114,92)(88,118,122,95)(89,123,127,103)(90,128,131,102)(91,132,134,109)(96,142,144,203)(97,145,146,205)(98,147,117,138)(101,129,154,196)(104,160,163,201)(105,164,162,150)(108,124,171,188)(110,172,156,167)(113,137,175,202)(119,179,181,152)(120,182,183,151)(121,141,135,177)(125,189,191,206)(126,159,176,186)(130,157,166,193)(133,178,198,140)(143,204,169,190)(148,185,199,155)(149,184,200,158)(153,197,165,195)(161,194,207,180)(170,192,173,187)

Conjugacy classes & character table

1435 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 90616);

Q128×D79 #34209317