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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
Abelian Cyclic Solvable Simple
Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Min. transitive degree Min. permutation degree
Number of subgroups Num. of normal subs Num. of conj. classes
Order statistics Family Order factorization Commutator length
Min. degree of $\C$-irrep Min. degree of $\Q$-irrep Min. degree of $\C$-rep Min. degree of $\Q$-rep
Outer aut. group Outer automorphisms Metabelian Metacyclic
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
Supersolvable Monomial Rational
Num. of char. subgroups Num. of aut. classes Number of divisions
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Results (1-50 of 365 matches)

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Label Name Family name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
2.1 $C_2$ $\ASigmaL(1,2)$ $2$ $2$ 2 $C_2$ $C_1$ $C_1$ $C_{2}$ Cyclic
3.1 $C_3$ $\ASigmaL(1,3)$ $3$ $3$ 3 $C_3$ $C_1$ $C_1$ $C_{3}$ Cyclic
5.1 $C_5$ $\ASigmaL(1,5)$ $5$ $5$ 5 $C_5$ $C_1$ $C_1$ $C_{5}$ Cyclic
7.1 $C_7$ $\ASigmaL(1,7)$ $7$ $7$ 7 $C_7$ $C_1$ $C_1$ $C_{7}$ Cyclic
8.3 $D_4$ $\ASigmaL(1,4)$ $2^{3}$ $2^{2}$ 5 $C_2$ $C_2^2$ $C_2$ $C_{2}^{2}$ Nilpotent - 2
11.1 $C_{11}$ $\ASigmaL(1,11)$ $11$ $11$ 11 $C_{11}$ $C_1$ $C_1$ $C_{11}$ Cyclic
13.1 $C_{13}$ $\ASigmaL(1,13)$ $13$ $13$ 13 $C_{13}$ $C_1$ $C_1$ $C_{13}$ Cyclic
17.1 $C_{17}$ $\ASigmaL(1,17)$ $17$ $17$ 17 $C_{17}$ $C_1$ $C_1$ $C_{17}$ Cyclic
18.3 $C_3\times S_3$ $\ASigmaL(1,9)$ $2 \cdot 3^{2}$ $2 \cdot 3$ 9 $C_3$ $S_3$ $C_3$ $C_{6}$ Solvable - 2
19.1 $C_{19}$ $\ASigmaL(1,19)$ $19$ $19$ 19 $C_{19}$ $C_1$ $C_1$ $C_{19}$ Cyclic
23.1 $C_{23}$ $\ASigmaL(1,23)$ $23$ $23$ 23 $C_{23}$ $C_1$ $C_1$ $C_{23}$ Cyclic
24.12 $S_4$ $\ASigmaL(2,2)$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ 5 $C_1$ $S_4$ $A_4$ $C_{2}$ Solvable - 3
24.13 $C_2\times A_4$ $\ASigmaL(1,8)$ $2^{3} \cdot 3$ $2 \cdot 3$ 8 $C_2$ $A_4$ $C_2^2$ $C_{6}$ Solvable - 2
29.1 $C_{29}$ $\ASigmaL(1,29)$ $29$ $29$ 29 $C_{29}$ $C_1$ $C_1$ $C_{29}$ Cyclic
31.1 $C_{31}$ $\ASigmaL(1,31)$ $31$ $31$ 31 $C_{31}$ $C_1$ $C_1$ $C_{31}$ Cyclic
37.1 $C_{37}$ $\ASigmaL(1,37)$ $37$ $37$ 37 $C_{37}$ $C_1$ $C_1$ $C_{37}$ Cyclic
41.1 $C_{41}$ $\ASigmaL(1,41)$ $41$ $41$ 41 $C_{41}$ $C_1$ $C_1$ $C_{41}$ Cyclic
43.1 $C_{43}$ $\ASigmaL(1,43)$ $43$ $43$ 43 $C_{43}$ $C_1$ $C_1$ $C_{43}$ Cyclic
47.1 $C_{47}$ $\ASigmaL(1,47)$ $47$ $47$ 47 $C_{47}$ $C_1$ $C_1$ $C_{47}$ Cyclic
50.3 $C_5\times D_5$ $\ASigmaL(1,25)$ $2 \cdot 5^{2}$ $2 \cdot 5$ 20 $C_5$ $D_5$ $C_5$ $C_{10}$ Solvable - 2
53.1 $C_{53}$ $\ASigmaL(1,53)$ $53$ $53$ 53 $C_{53}$ $C_1$ $C_1$ $C_{53}$ Cyclic
59.1 $C_{59}$ $\ASigmaL(1,59)$ $59$ $59$ 59 $C_{59}$ $C_1$ $C_1$ $C_{59}$ Cyclic
61.1 $C_{61}$ $\ASigmaL(1,61)$ $61$ $61$ 61 $C_{61}$ $C_1$ $C_1$ $C_{61}$ Cyclic
64.32 $C_2\wr C_4$ $\ASigmaL(1,16)$ $2^{6}$ $2^{3}$ 13 $C_2$ $C_2^3:C_4$ $C_2^3$ $C_{2} \times C_{4}$ Nilpotent - 4
67.1 $C_{67}$ $\ASigmaL(1,67)$ $67$ $67$ 67 $C_{67}$ $C_1$ $C_1$ $C_{67}$ Cyclic
71.1 $C_{71}$ $\ASigmaL(1,71)$ $71$ $71$ 71 $C_{71}$ $C_1$ $C_1$ $C_{71}$ Cyclic
73.1 $C_{73}$ $\ASigmaL(1,73)$ $73$ $73$ 73 $C_{73}$ $C_1$ $C_1$ $C_{73}$ Cyclic
79.1 $C_{79}$ $\ASigmaL(1,79)$ $79$ $79$ 79 $C_{79}$ $C_1$ $C_1$ $C_{79}$ Cyclic
81.7 $C_3\wr C_3$ $\ASigmaL(1,27)$ $3^{4}$ $3^{2}$ 17 $C_3$ $\He_3$ $C_3^2$ $C_{3}^{2}$ Nilpotent - 3
83.1 $C_{83}$ $\ASigmaL(1,83)$ $83$ $83$ 83 $C_{83}$ $C_1$ $C_1$ $C_{83}$ Cyclic
89.1 $C_{89}$ $\ASigmaL(1,89)$ $89$ $89$ 89 $C_{89}$ $C_1$ $C_1$ $C_{89}$ Cyclic
97.1 $C_{97}$ $\ASigmaL(1,97)$ $97$ $97$ 97 $C_{97}$ $C_1$ $C_1$ $C_{97}$ Cyclic
98.3 $C_7\times D_7$ $\ASigmaL(1,49)$ $2 \cdot 7^{2}$ $2 \cdot 7$ 35 $C_7$ $D_7$ $C_7$ $C_{14}$ Solvable - 2
101.1 $C_{101}$ $\ASigmaL(1,101)$ $101$ $101$ 101 $C_{101}$ $C_1$ $C_1$ $C_{101}$ Cyclic
103.1 $C_{103}$ $\ASigmaL(1,103)$ $103$ $103$ 103 $C_{103}$ $C_1$ $C_1$ $C_{103}$ Cyclic
107.1 $C_{107}$ $\ASigmaL(1,107)$ $107$ $107$ 107 $C_{107}$ $C_1$ $C_1$ $C_{107}$ Cyclic
109.1 $C_{109}$ $\ASigmaL(1,109)$ $109$ $109$ 109 $C_{109}$ $C_1$ $C_1$ $C_{109}$ Cyclic
113.1 $C_{113}$ $\ASigmaL(1,113)$ $113$ $113$ 113 $C_{113}$ $C_1$ $C_1$ $C_{113}$ Cyclic
127.1 $C_{127}$ $\ASigmaL(1,127)$ $127$ $127$ 127 $C_{127}$ $C_1$ $C_1$ $C_{127}$ Cyclic
131.1 $C_{131}$ $\ASigmaL(1,131)$ $131$ $131$ 131 $C_{131}$ $C_1$ $C_1$ $C_{131}$ Cyclic
137.1 $C_{137}$ $\ASigmaL(1,137)$ $137$ $137$ 137 $C_{137}$ $C_1$ $C_1$ $C_{137}$ Cyclic
139.1 $C_{139}$ $\ASigmaL(1,139)$ $139$ $139$ 139 $C_{139}$ $C_1$ $C_1$ $C_{139}$ Cyclic
149.1 $C_{149}$ $\ASigmaL(1,149)$ $149$ $149$ 149 $C_{149}$ $C_1$ $C_1$ $C_{149}$ Cyclic
151.1 $C_{151}$ $\ASigmaL(1,151)$ $151$ $151$ 151 $C_{151}$ $C_1$ $C_1$ $C_{151}$ Cyclic
157.1 $C_{157}$ $\ASigmaL(1,157)$ $157$ $157$ 157 $C_{157}$ $C_1$ $C_1$ $C_{157}$ Cyclic
160.235 $C_2\wr C_5$ $\ASigmaL(1,32)$ $2^{5} \cdot 5$ $2 \cdot 5$ 16 $C_2$ $C_2^4:C_5$ $C_2^4$ $C_{10}$ Solvable - 2
163.1 $C_{163}$ $\ASigmaL(1,163)$ $163$ $163$ 163 $C_{163}$ $C_1$ $C_1$ $C_{163}$ Cyclic
167.1 $C_{167}$ $\ASigmaL(1,167)$ $167$ $167$ 167 $C_{167}$ $C_1$ $C_1$ $C_{167}$ Cyclic
173.1 $C_{173}$ $\ASigmaL(1,173)$ $173$ $173$ 173 $C_{173}$ $C_1$ $C_1$ $C_{173}$ Cyclic
179.1 $C_{179}$ $\ASigmaL(1,179)$ $179$ $179$ 179 $C_{179}$ $C_1$ $C_1$ $C_{179}$ Cyclic
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