#34176083

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×M₁₂₈ ⋊ C₂
Sylow	C₇₉ ⋊ #7514

Subgroups

	subgroup	G/H
center	C₃₂	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×M₁₂₈	C₂
socle	C₂×C₇₉	C₂²⋊C₃₂
prime radical	#11929624	C₁₆
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7514
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 207 points:

(1,2)(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)(80,82)(81,89)(83,102)(84,106)(85,99)(86,100)(87,97)(88,101)(90,121)(91,125)(92,118)(93,119)(94,96)(95,120)(98,110)(103,138)(104,146)(105,147)(107,152)(108,153)(109,116)(111,143)(112,139)(113,144)(114,140)(115,145)(117,129)(122,135)(123,168)(124,169)(126,173)(127,174)(128,151)(130,165)(131,136)(132,166)(133,137)(134,167)(141,156)(142,158)(148,185)(149,186)(150,190)(154,194)(155,161)(157,162)(159,187)(160,189)(163,177)(164,179)(170,182)(171,183)(172,200)(175,202)(176,192)(178,193)(180,184)(181,199)(188,196)(191,206)(195,197)(198,204)(201,205)(203,207);(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)(80,81,84,91,85,92,107,126,86,93,108,127,111,130,154,175,87,94,109,128,112,131,155,176,114,133,157,178,159,180,195,203,88,95,110,129,113,132,156,177,115,134,158,179,160,181,196,204,83,90,103,122,104,123,148,170,105,124,149,171,150,172,191,201)(82,96,98,135,99,136,141,182,100,137,142,183,143,184,188,205,97,120,138,125,139,166,185,173,140,167,186,174,187,199,206,202,101,121,106,151,144,168,152,192,145,169,153,193,189,200,194,207,102,89,116,117,146,118,161,163,147,119,162,164,190,165,197,198)

Polycyclic presentation:

2^8,79|a^2=d,c^2=h,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bc,i^a=i^78,c^b=ch,d^b=dh,i^b=i^78

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 57382);