Finite Group DatabasebetaFinite Group Databasebeta| order | 33450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C25⋊(D223⋊C3) |
| unambiguous semidirect product | (C223⋊C3)⋊D25 |
| unambiguous semidirect product | D5575⋊C3 |
| Fitting, derived, Hall | C25×C223 ⋊ C2×C3 |
| Hall | C25×C223⋊C3 ⋊ C2 |
| Hall | D5575 ⋊ C3 |
| Hall | C223⋊C3 ⋊ D25 |
| Sylow | C223 ⋊ C3×D25 |
| Sylow | C25 ⋊ D223⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C25×C223 | C2×C3 |
| Frattini | C5 | C5⋊(D223⋊C3) |
| socle | C5×C223 | C3×D5 |
| LCS | G ▷ C25×C223 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 248 |
| transitive degree | μt(G) = 5575 |
A minimal faithful permutation representation — 2 generators on 248 points:
(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(27,31,32,28,29,33)(30,39,40,34,35,49)(36,52,53,42,43,66)(37,51,47,45,41,38)(44,71,72,55,56,89)(46,69,70,58,54,63)(48,50,68,60,61,64)(57,94,95,74,75,118)(59,92,93,77,73,86)(62,67,91,79,80,87)(65,107,105,83,84,82)(76,123,124,97,98,153)(78,121,122,100,96,115)(81,90,120,102,103,116)(85,136,137,108,109,111)(88,142,134,112,113,106)(99,158,159,126,127,194)(101,156,157,129,125,150)(104,119,155,131,132,151)(110,171,172,138,139,146)(114,178,176,143,144,141)(117,183,169,147,148,135)(128,199,200,161,162,233)(130,197,198,164,160,191)(133,154,196,166,167,192)(140,210,211,173,174,187)(145,217,218,179,180,182)(149,222,215,184,185,177)(152,226,208,188,189,170)(163,212,213,202,175,240)(165,235,236,203,201,231)(168,195,234,205,206,232)(181,244,207,219,220,225)(186,245,239,223,224,221)(190,246,243,227,228,216)(193,247,238,229,230,209)(204,241,242,237,214,248);(1,15,25,18,4,7,21,19,10,13,16,14,12,11,20,8,6,5,24,17,3,2,22,23,9)(26,27,30,36,44,57,76,99,128,163,198,124,155,183,178,171,156,123,154,193,227,188,132,75,96,109,113,116,118,150,187,225,205,166,131,102,79,60,47,32,31,37,46,59,78,101,130,165,204,236,159,196,226,222,217,210,197,158,195,223,184,147,103,56,73,84,87,89,115,146,182,221,238,208,169,134,105,70,40,51,39,50,65,85,110,140,175,214,242,200,234,247,246,245,244,240,235,199,219,179,143,112,80,43,54,64,66,86,111,141,177,216,243,215,176,137,93,53,68,69,52,67,88,114,145,181,162,201,213,220,224,228,230,232,233,248,241,212,173,138,108,83,61,35,41,49,63,82,106,135,170,209,239,218,172,122,72,91,107,92,71,90,117,149,186,206,127,160,174,180,185,189,192,194,231,237,203,164,129,100,77,58,45,29,33,38,48,62,81,104,133,168,207,211,157,95,120,142,136,121,94,119,152,190,229,167,98,125,139,144,148,151,153,191,202,161,126,97,74,55,42,34,28)Polycyclic presentation:
2,3,5,223,5|c^5=e^4,c^a=c^4e,d^a=d^222,e^a=e^4,d^b=d^183967 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33450, 12);