Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C2×2−1+4×C101 ⋊ C2 |
| Sylow | C101 ⋊ 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D101 |
| commutator, socle, cosocle | C2×C101 | C26 |
| Fitting | C2×2−1+4×C101 | C2 |
| hypercenter | C2×2−1+4 | D101 |
| hypocenter | C101 | 2−1+6 |
| LCS | G ▷ C2×C101 ▷ C101 | |
| UCS | C1 ◁ C2 ◁ C2×2−1+4 | |
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 6 generators on 133 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)(103,111)(108,112)(109,123)(110,125)(120,126)(121,127)(122,124)(132,133);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(104,115)(108,112)(113,129)(114,131)(120,126)(121,127)(128,130)(132,133);(102,106,107,119)(103,110,111,125)(104,114,115,131)(105,116,117,118)(108,121,112,127)(109,122,123,124)(113,128,129,130)(120,132,126,133);(102,105,107,117)(103,109,111,123)(104,113,115,129)(106,118,119,116)(108,120,112,126)(110,124,125,122)(114,130,131,128)(121,133,127,132);(102,104)(103,108)(105,113)(106,114)(107,115)(109,120)(110,121)(111,112)(116,128)(117,129)(118,130)(119,131)(122,132)(123,126)(124,133)(125,127);(102,103)(104,112)(105,109)(106,110)(107,111)(108,115)(113,126)(114,127)(116,122)(117,123)(118,124)(119,125)(120,129)(121,131)(128,133)(130,132)Polycyclic presentation:
2^7,101|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^f=h^1001765 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 19322);