Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C97 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20260570 |
| commutator, cosocle | C32×C97 | C22 |
| Frattini | C32 | C2×D97 |
| Fitting | Q64×C97 | C2 |
| socle, prime radical | C2×C97 | D32 |
| hypercenter | Q64 | D97 |
| hypocenter | C97 | Q128 |
| LCS | G ▷ C32×C97 ▷ C16×C97 ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 225 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 2 generators on 225 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)(98,100,105,117)(99,106,111,140)(101,119,124,172)(102,126,129,189)(103,131,133,193)(104,135,132,116)(107,142,145,206)(108,147,149,214)(109,151,152,216)(110,153,134,139)(112,154,157,118)(113,141,136,121)(114,146,162,120)(115,150,164,127)(122,179,181,225)(123,183,180,171)(125,137,186,199)(128,191,174,188)(130,138,192,201)(143,208,209,222)(144,210,194,205)(148,215,184,213)(155,219,220,178)(156,168,165,182)(158,207,196,224)(159,177,195,204)(160,217,221,166)(161,175,187,211)(163,218,223,167)(169,203,200,173)(170,202,197,176)(185,212,190,198);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(98,99,105,111)(100,112,117,157)(101,118,124,154)(102,125,129,186)(103,130,133,192)(104,134,132,110)(106,136,140,113)(107,141,145,121)(108,146,149,120)(109,150,152,127)(114,160,162,221)(115,163,164,223)(116,165,135,156)(119,147,172,214)(122,178,181,219)(123,182,180,168)(126,142,189,206)(128,190,174,185)(131,155,193,220)(137,197,199,170)(138,200,201,169)(139,159,153,195)(143,207,209,224)(144,177,194,204)(148,175,184,211)(151,196,216,158)(161,222,187,208)(166,203,217,173)(167,202,218,176)(171,215,183,213)(179,212,225,198)(188,210,191,205)Polycyclic presentation:
2^7,97|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^96947 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 2727);