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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central
Ambient label Ambient order Direct product Semidirect product
Quotient label Subgroup index Cyclic quotient Abelian quotient Solvable quotient
Maximal quotient Non-trivial proper Hall subgroup Sylow subgroup
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Results (9 matches)

  displayed columns for results
Label Subgroup Ambient Quotient
Name Order Sylow norm char max cent ab Name Order Name Size max ab
9072.b.1.a1.a1 $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $C_1$ $1$ ✓
9072.b.2.a1.a1 $C_3\times {}^2G(2,3)$ $2^{3} \cdot 3^{4} \cdot 7$ ✓ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $C_2$ $2$ ✓
9072.b.3.a1.a1 $S_3\times \SL(2,8)$ $2^{4} \cdot 3^{3} \cdot 7$ ✓ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $C_3$ $3$ ✓
9072.b.6.a1.a1 $C_3\times \SL(2,8)$ $2^{3} \cdot 3^{3} \cdot 7$ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $C_6$ $2 \cdot 3$ ✓
9072.b.6.b1.a1 ${}^2G(2,3)$ $2^{3} \cdot 3^{3} \cdot 7$ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $S_3$ $2 \cdot 3$
9072.b.18.a1.a1 $\SL(2,8)$ $2^{3} \cdot 3^{2} \cdot 7$ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $C_3\times S_3$ $2 \cdot 3^{2}$ ✓
9072.b.1512.a1.a1 $S_3$ $2 \cdot 3$ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ ${}^2G(2,3)$ $2^{3} \cdot 3^{3} \cdot 7$
9072.b.3024.a1.a1 $C_3$ $3$ ✓ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $\SL(2,8):C_6$ $2^{4} \cdot 3^{3} \cdot 7$ ✓
9072.b.9072.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$ $S_3\times {}^2G(2,3)$ $2^{4} \cdot 3^{4} \cdot 7$
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