Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C79×D64 ⋊ C2 |
| Sylow | C79 ⋊ D128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #23761612 |
| commutator, cosocle | C64×C79 | C22 |
| Frattini | C64 | C2×D79 |
| Fitting | C79×D64 | C2 |
| socle | C2×C79 | D64 |
| hypercenter | D64 | D79 |
| hypocenter | C79 | D128 |
| LCS | G ▷ C64×C79 ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ D64 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(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)(82,95)(83,96)(84,102)(85,107)(86,111)(89,90)(91,126)(92,129)(93,131)(94,118)(97,141)(98,145)(99,148)(100,136)(101,114)(103,157)(104,161)(105,140)(106,115)(108,167)(109,155)(110,116)(112,165)(113,117)(119,150)(120,149)(121,171)(122,125)(123,168)(124,170)(127,185)(128,184)(130,186)(132,133)(134,142)(135,146)(137,138)(139,158)(143,200)(144,198)(147,201)(151,187)(152,174)(153,154)(156,172)(159,206)(160,173)(162,183)(163,182)(164,176)(166,175)(169,177)(178,205)(179,204)(180,203)(181,207)(188,197)(189,196)(190,192)(191,202)(193,199)(194,195);(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)(82,88)(83,101)(84,106)(85,110)(86,113)(87,94)(89,95)(90,96)(91,102)(92,107)(93,111)(97,126)(98,129)(99,131)(100,118)(103,156)(104,160)(105,152)(108,166)(109,164)(112,169)(114,133)(115,132)(116,142)(117,146)(119,141)(120,145)(121,148)(122,136)(123,157)(124,161)(125,140)(127,167)(128,155)(130,165)(134,168)(135,170)(137,150)(138,149)(139,171)(143,185)(144,184)(147,186)(151,158)(153,163)(154,162)(159,205)(172,197)(173,196)(174,192)(175,202)(176,190)(177,199)(178,200)(179,198)(180,201)(181,206)(182,189)(183,188)(187,191)(193,207)(194,204)(195,203)Polycyclic presentation:
2^8,79|c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^781432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57480);