Finite Group DatabasebetaFinite Group Databasebeta| order | 13184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C103 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20267626 |
| commutator, cosocle | C32×C103 | C22 |
| Frattini | C32 | C2×D103 |
| Fitting | Q64×C103 | C2 |
| socle, prime radical | C2×C103 | D32 |
| hypercenter | Q64 | D103 |
| hypocenter | C103 | Q128 |
| LCS | G ▷ C32×C103 ▷ C16×C103 ▷ C8×C103 ▷ C4×C103 ▷ C2×C103 ▷ C103 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 231 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 2 generators on 231 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)(102,103)(104,106,111,123)(105,112,117,146)(107,125,130,178)(108,132,135,195)(109,137,139,199)(110,141,138,122)(113,148,151,212)(114,153,155,220)(115,157,158,222)(116,159,140,145)(118,160,163,124)(119,147,142,127)(120,152,168,126)(121,156,170,133)(128,185,187,231)(129,189,186,177)(131,143,192,205)(134,197,180,194)(136,144,198,207)(149,214,215,228)(150,216,200,211)(154,221,190,219)(161,225,226,184)(162,174,171,188)(164,213,202,230)(165,183,201,210)(166,223,227,172)(167,181,193,217)(169,224,229,173)(175,209,206,179)(176,208,203,182)(191,218,196,204);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(104,105,111,117)(106,118,123,163)(107,124,130,160)(108,131,135,192)(109,136,139,198)(110,140,138,116)(112,142,146,119)(113,147,151,127)(114,152,155,126)(115,156,158,133)(120,166,168,227)(121,169,170,229)(122,171,141,162)(125,153,178,220)(128,184,187,225)(129,188,186,174)(132,148,195,212)(134,196,180,191)(137,161,199,226)(143,203,205,176)(144,206,207,175)(145,165,159,201)(149,213,215,230)(150,183,200,210)(154,181,190,217)(157,202,222,164)(167,228,193,214)(172,209,223,179)(173,208,224,182)(177,221,189,219)(185,218,231,204)(194,216,197,211)Polycyclic presentation:
2^7,103|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^1021004 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13184, 2727);