#34176179

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₇₉×C₁₂₈ ⋊ C₂
Sylow	C₇₉ ⋊ M₂₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₇₉
commutator, socle	C₂×C₇₉	C₂×C₆₄
Fitting	C₇₉×C₁₂₈	C₂
prime radical	C₂×D₇₉	C₆₄
cosocle	C₆₄×C₇₉	C₂²
hypercenter	C₁₂₈	D₇₉
hypocenter	C₇₉	M₂₅₆
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

44 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:

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

Polycyclic presentation:

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

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, 57478);