Finite Group DatabasebetaFinite Group Databasebeta| order | 23034 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C11⋊(D349⋊C3) |
| unambiguous semidirect product | (C349⋊C3)⋊D11 |
| unambiguous semidirect product | D3839⋊C3 |
| Fitting, socle, derived, Hall | C11×C349 ⋊ C2×C3 |
| Hall | C11×C349⋊C3 ⋊ C2 |
| Hall | D3839 ⋊ C3 |
| Hall | C349⋊C3 ⋊ D11 |
| Sylow | C349 ⋊ C3×D11 |
| Sylow | C11 ⋊ D349⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C11×C349 | C2×C3 |
| LCS | G ▷ C11×C349 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 360 |
| transitive degree | μt(G) = 3839 |
A minimal faithful permutation representation — 2 generators on 360 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,268)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,273,274,224,225,313)(181,271,272,227,223,265)(184,217,270,229,230,266)(191,280,281,236,237,262)(198,285,286,243,244,257)(203,292,293,249,250,252)(208,298,290,254,255,247)(212,303,226,259,260,240)(215,308,279,263,264,233)(228,314,315,276,275,311)(231,269,310,278,267,312)(238,316,317,282,283,309)(245,320,321,287,288,307)(251,325,326,294,295,302)(256,330,319,299,300,297)(261,335,277,304,305,291)(284,336,337,318,306,340)(289,341,342,322,323,339)(296,343,344,327,328,338)(301,346,324,331,332,334)(329,347,348,345,333,349)(351,352)(353,354)(355,356)(357,358)(359,360);(1,2,5,11,19,32,51,74,103,138,179,226,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,289,324,326,286,235,173,99,130,158,153,146,131,98,129,168,212,261,306,323,328,332,337,315,222,270,308,303,298,292,285,280,271,221,269,230,137,176,190,197,202,207,211,214,216,265,309,339,349,347,331,299,254,206,163,107,50,71,84,88,91,93,125,162,204,252,297,318,282,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,278,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,277,290,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,296,329,344,321,281,220,134,171,205,200,194,187,172,133,170,215,225,275,283,288,295,300,305,274,310,312,313,335,330,325,320,316,314,273,263,183,102,135,149,155,160,164,167,169,213,262,307,338,345,327,294,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,291,276,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,279,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,284,319,293,242,188,132,70,95,117,111,96,69,94,127,165,208,256,301,333,348,342,317,272,175,218,258,253,248,241,234,219,174,217,267,178,223,237,244,250,255,260,264,266,268,311,340,346,343,341,336,304,259,210,142,73,100,114,119,123,126,128,166,209,257,302,334,322,287,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,224,177,136,101,72,49,30,17,9,3)(350,351,353,355,357,359,360,358,356,354,352)Polycyclic presentation:
2,3,11,349|c^a=c^10,d^a=d^348,d^b=d^226659 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23034, 12);