Q₆₄×D₁₅₇ #34102439

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

direct product

Q₆₄×D₁₅₇

Splittings

Fitting	Q₆₄×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ C₂×Q₆₄

Subgroups

	subgroup	G/H
center	C₂	D₁₆×D₁₅₇
commutator, cosocle	C₁₆×C₁₅₇	C₂³
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	Q₆₄×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂×D₁₆
prime radical	C₂×D₁₅₇	D₁₆
hypercenter	Q₆₄	D₁₅₇
hypocenter	C₁₅₇	C₂×Q₆₄
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 221
transitive degree	μ_t(G) = 10048

A minimal faithful permutation representation — 3 generators on 221 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)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147)(148,149)(150,151)(152,153)(154,155)(156,157);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150,152,154,156,157,155,153,151,149,147,145,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(158,160,164,174)(159,165,169,190)(161,176,180,210)(162,182,184,216)(163,186,183,173)(166,192,194,219)(167,196,197,221)(168,198,185,189)(170,199,201,175)(171,191,187,178)(172,195,204,177)(179,214,212,209)(181,188,215,217)(193,220,206,203)(200,207,205,213)(202,211,208,218);(158,159,164,169)(160,170,174,201)(161,175,180,199)(162,181,184,215)(163,185,183,168)(165,187,190,171)(166,191,194,178)(167,195,197,177)(172,203,204,220)(173,205,186,200)(176,196,210,221)(179,213,212,207)(182,192,216,219)(188,214,217,209)(189,202,198,208)(193,211,206,218)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 6813);

Q64×D157 #34102439