Finite Group DatabasebetaFinite Group Databasebeta| order | 2944 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C23 · Schur multiplier (H2) = C4 |
| direct product | C23×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C23 | C2×D4 |
| commutator | C2×C4 | C22×C4×C23 |
| Frattini, cosocle | C42 | C23×C23 |
| socle | C22×C23 | C2×C4⋊C4 |
| prime radical | C23×C23 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G | |
150 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1472 |
A minimal faithful permutation representation — 3 generators on 55 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)(26,33)(27,31)(28,37)(29,32)(30,35)(34,38)(36,39)(40,41,43,46,44,47,50,53)(42,45,49,52,48,51,54,55);(24,26,28,35,29,36,27,34)(25,30,31,33,32,38,37,39)(40,42)(41,45)(43,49)(44,48)(46,52)(47,51)(50,54)(53,55);(24,27,29,28)(25,31,32,37)(26,34,36,35)(30,33,38,39)(42,48)(45,51)(49,54)(52,55)874 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C23×#1572, C23×#17803, C23×#17808, … |
| maximal normal subgroup of | C2×C23×#1572, C23×#17803, C23×#17808, … |
| maximal quotient of | C23×#7878, C23×#8277, C23×#8278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2944, 304);