Properties

Label ab/6.468
Order \( 2^{3} \cdot 3^{3} \cdot 13 \)
Exponent \( 2^{2} \cdot 3^{2} \cdot 13 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{7} \cdot 3^{4} \)
Trans deg. $2808$
Rank $2$

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This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.

Group information

Description:$C_{6} \times C_{468}$
Order: \(2808\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 13 \)
Exponent: \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \)
Automorphism group:Group of order 10368
Nilpotency class:$1$
Derived length:$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group). Whether it is metacyclic or rational has not been computed.

Group statistics

Order 1 2 3 4 6 9 12 13 18 26 36 39 52 78 117 156 234 468
Elements 1 3 8 4 24 18 32 12 54 36 72 96 48 288 216 384 648 864 2808
Conjugacy classes   1 3 8 4 24 18 32 12 54 36 72 96 48 288 216 384 648 864 2808
Divisions data not computed
Autjugacy classes data not computed

Dimension 1
Irr. complex chars.   2808 2808

Constructions

Rank: $2$
Inequivalent generating pairs: not computed

Homology

Primary decomposition: $C_{2} \times C_{4} \times C_{3} \times C_{9} \times C_{13}$

Subgroups

Center: $Z \simeq$ $C_{6} \times C_{468}$ $G/Z \simeq$ $C_1$
Commutator: $G' \simeq$ $C_1$ $G/G' \simeq$ $C_{6} \times C_{468}$
Frattini: $\Phi \simeq$ $C_6$ $G/\Phi \simeq$ $C_6\times C_{78}$
Fitting: $\operatorname{Fit} \simeq$ $C_{6} \times C_{468}$ $G/\operatorname{Fit} \simeq$ $C_1$
Radical: $R \simeq$ $C_{6} \times C_{468}$ $G/R \simeq$ $C_1$
Socle: $S \simeq$ $C_6\times C_{78}$ $G/S \simeq$ $C_6$
2-Sylow subgroup: $P_{2} \simeq$ $C_2\times C_4$
3-Sylow subgroup: $P_{3} \simeq$ $C_3\times C_9$
13-Sylow subgroup: $P_{13} \simeq$ $C_{13}$