Abstract group 2413320.a: $\PSL(2,169)$

• Label: 2413320.a
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $170$
• Trans deg.: $170$
• Rank: $2$

Group information

Description:	$\PSL(2,169)$
Order:	\(2413320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \)
Exponent:	\(92820\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
Automorphism group:	$\PSL(2,169).C_2^2$, of order \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \)
Composition factors:	$\PSL(2,169)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	12	13	14	17	21	28	42	84	85
Elements	1	14365	28730	28730	56784	28730	86190	57460	28560	86190	227136	172380	172380	172380	344760	908544	2413320
Conjugacy classes	1	1	1	1	2	1	3	2	2	3	8	6	6	6	12	32	87
Divisions	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	17
Autjugacy classes	1	1	1	1	1	1	3	2	1	3	4	3	3	3	6	16	50

Dimension	1	85	168	169	170	336	340	510	1020	1344	2040	5376
Irr. complex chars.	1	2	42	1	41	0	0	0	0	0	0	0	87
Irr. rational chars.	1	2	0	1	3	1	1	2	3	1	1	1	17

Minimal presentations

Permutation degree:	$170$
Transitive degree:	$170$
Rank:	$2$
Inequivalent generating pairs:	$598703$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	85	85	85
Arbitrary	85	85	85

Constructions

Groups of Lie type:	$\PSL(2,169)$, $\PSU(2,169)$, $\Omega(3,169)$, $\OmegaMinus(4,13)$, $\PSOMinus(4,13)$, $\POmega(3,169)$, $\POmegaMinus(4,13)$
Permutation group:	Degree $170$ $\langle(1,2,3)(4,7,8)(5,9,11)(6,13,14)(10,21,23)(12,27,28)(15,34,35)(16,36,38) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,169)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 2782197 subgroups in 71 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(2,169)$
Commutator:	$G' \simeq$ $\PSL(2,169)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(2,169)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(2,169)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(2,169)$
Socle:	$\operatorname{soc} \simeq$ $\PSL(2,169)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}^2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 2413320: $\PSL(2,169)$

Order 14196: $C_{13}^2:C_{84}$

Order 7098: $C_{13}^2:C_{42}$

Order 4732: $C_{13}^2:C_{28}$

Order 3549: $C_{13}^2:C_{21}$

Order 2366: $C_{13}^2:C_{14}$

Order 2184: $\PGL(2,13)$ x 2

Order 2028: $C_{13}:F_{13}$

Order 1183: $C_{13}^2:C_7$

Order 1092: $\PSL(2,13)$ x 2

Order 1014: $C_{13}^2:C_6$

Order 676: $C_{13}^2:C_4$

Order 507: $C_{13}^2:C_3$

Order 338: $C_{13}:D_{13}$

Order 170: $D_{85}$

Order 169: $C_{13}^2$

Order 168: $D_{84}$

Order 156: $F_{13}$ x 2

Order 85: $C_{85}$

Order 84: $D_{42}$ x 2, $C_{84}$

Order 78: $C_{13}:C_6$ x 2

Order 60: $A_5$ x 2

Order 56: $D_{28}$

Order 52: $C_{13}:C_4$ x 2

Order 42: $D_{21}$ x 2, $C_{42}$

Order 39: $C_{13}:C_3$ x 2

Order 34: $D_{17}$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 26: $D_{13}$ x 2

Order 24: $S_4$ x 2, $D_{12}$

Order 21: $C_{21}$

Order 17: $C_{17}$

Order 14: $D_7$ x 2, $C_{14}$

Order 13: $C_{13}$ x 2

Order 12: $D_6$ x 2, $A_4$ x 2, $C_{12}$

Order 10: $D_5$

Order 8: $D_4$

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2413320: $\PSL(2,169)$

Order 14196: $C_{13}^2:C_{84}$

Order 7098: $C_{13}^2:C_{42}$

Order 4732: $C_{13}^2:C_{28}$

Order 3549: $C_{13}^2:C_{21}$

Order 2366: $C_{13}^2:C_{14}$

Order 2184: $\PGL(2,13)$

Order 2028: $C_{13}:F_{13}$

Order 1183: $C_{13}^2:C_7$

Order 1014: $C_{13}^2:C_6$

Order 676: $C_{13}^2:C_4$

Order 507: $C_{13}^2:C_3$

Order 338: $C_{13}:D_{13}$

Order 170: $D_{85}$

Order 169: $C_{13}^2$

Order 168: $D_{84}$

Order 85: $C_{85}$

Order 56: $D_{28}$

Order 34: $D_{17}$

Order 21: $C_{21}$

Order 17: $C_{17}$

Order 10: $D_5$

Order 8: $D_4$

Order 7: $C_7$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2413320: $\PSL(2,169)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,169)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2413320: $\PSL(2,169)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,169)$)

Series

Derived series	$\PSL(2,169)$
Chief series	$\PSL(2,169)$	$\rhd$	$C_1$
Lower central series	$\PSL(2,169)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $87 \times 87$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.