Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×#2262 ⋊ C2 |
| Sylow | C79 ⋊ #13742 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#19098398 |
| commutator, cosocle | C32×C79 | C23 |
| Frattini | C32 | C22×D79 |
| Fitting | C79×#2262 | C2 |
| socle | C2×C79 | C2×D32 |
| hypercenter | #2262 | D79 |
| hypocenter | C79 | #13742 |
| LCS | G ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ #2262 | |
134 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(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)(81,88)(87,106)(89,110)(90,111)(91,112)(92,105)(98,129)(102,135)(107,137)(108,138)(109,117)(113,139)(114,136)(115,130)(116,140)(131,133);(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)(81,87)(82,93)(83,94)(84,99)(85,103)(88,106)(89,90)(91,116)(92,117)(95,125)(96,128)(97,121)(98,107)(100,134)(101,124)(102,108)(104,132)(105,109)(110,111)(112,140)(113,131)(114,130)(115,136)(118,119)(120,126)(122,123)(127,143)(129,137)(133,139)(135,138)(141,142);(80,81)(82,87)(83,98)(84,102)(85,105)(86,88)(89,93)(90,94)(91,99)(92,103)(95,116)(96,117)(97,106)(100,133)(101,129)(104,135)(107,119)(108,118)(109,126)(110,121)(111,124)(112,132)(113,125)(114,128)(115,134)(120,136)(122,131)(123,130)(127,140)(137,142)(138,141)(139,143)Polycyclic presentation:
2^8,79|d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^781528 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90625);