Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2193 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C29 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C29×#131 ◁ G | |
210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 61 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)(31,37)(32,41)(33,44)(34,35)(38,52)(39,56)(40,51)(42,58)(43,48)(45,49)(46,47)(50,53)(54,61)(55,59);(30,31)(32,38)(33,45)(34,39)(35,49)(36,40)(37,44)(41,53)(42,59)(43,55)(46,51)(47,56)(48,54)(50,57)(52,58)(60,61);(30,32,36,43)(31,38,40,55)(33,42,47,58)(34,48,35,41)(37,50,51,61)(39,54,49,53)(44,57,46,60)(45,59,56,52)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2193, C29×#33538, C29×#33540, … |
| maximal normal subgroup of | C2×C29×#2193, C29×#33538, C29×#33540, … |
| maximal quotient of | C29×#12109, C29×#12110, C29×#12165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 925);