Abstract group 39609.342: $C_{4401}.C_9$

• Label: 39609.342
• Order: \( 3^{5} \cdot 163 \)
• Exponent: \( 3^{4} \cdot 163 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{5} \)
• $\card{Z(G)}$: \( 3^{3} \)
• $\card{\Aut(G)}$: \( 2 \cdot 3^{8} \cdot 163 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{6} \)
• Perm deg.: $244$
• Trans deg.: $13203$
• Rank: $2$

Group information

Description:	$C_{4401}.C_9$
Order:	\(39609\)\(\medspace = 3^{5} \cdot 163 \)
Exponent:	\(13203\)\(\medspace = 3^{4} \cdot 163 \)
Automorphism group:	$C_{11}:(C_4\times F_{11})$, of order \(2138886\)\(\medspace = 2 \cdot 3^{8} \cdot 163 \)
Composition factors:	$C_3$ x 5, $C_{163}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.

Group statistics

Order	1	3	9	27	81	163	489	1467	4401
Elements	1	980	1962	5886	26406	162	324	972	2916	39609
Conjugacy classes	1	8	18	54	162	18	36	108	324	729
Divisions	1	4	3	3	3	1	1	1	1	18
Autjugacy classes	1	4	6	6	6	1	2	2	2	30

Minimal presentations

Permutation degree:	$244$
Transitive degree:	$13203$
Rank:	$2$
Inequivalent generating pairs:	$432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{4401}=1, a^{9}=b^{3912}, b^{a}=b^{379} \rangle$
Permutation group:	Degree $244$ $\langle(2,4,14,6,16,19,5,15,8)(3,9,29,11,31,34,10,30,12)(7,22,21,24,62,59,23,61,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 104 & 0 \\ 0 & 58 \end{array}\right), \left(\begin{array}{rr} 46 & 0 \\ 0 & 156 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{163})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{163}:C_{81})$ $\,\rtimes\,$ $C_3$ (3)	$(C_{163}:C_3)$ $\,\rtimes\,$ $C_{81}$ (3)	$C_{163}$ $\,\rtimes\,$ $(C_3\times C_{81})$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{4401}$ . $C_9$	$C_9$ . $(C_{489}:C_9)$	$(C_{4401}:C_3)$ . $C_3$	$(C_{1467}:C_3)$ . $C_9$	all 12

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{3} \times C_{81} $
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 2304 subgroups in 36 conjugacy classes, 22 normal (16 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{27}$	$G/Z \simeq$ $C_{163}:C_9$
Commutator:	$G' \simeq$ $C_{163}$	$G/G' \simeq$ $C_3\times C_{81}$
Frattini:	$\Phi \simeq$ $C_9$	$G/\Phi \simeq$ $C_{489}:C_9$
Fitting:	$\operatorname{Fit} \simeq$ $C_{4401}$	$G/\operatorname{Fit} \simeq$ $C_9$
Radical:	$R \simeq$ $C_{4401}.C_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{489}$	$G/\operatorname{soc} \simeq$ $C_3\times C_{27}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_{81}$
163-Sylow subgroup:	$P_{ 163 } \simeq$ $C_{163}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 39609: $C_{4401}.C_9$

Order 13203: $C_{163}:C_{81}$ x 3, $C_{4401}:C_3$

Order 4401: $C_{163}:C_{27}$ x 2, $C_{1467}:C_3$, $C_{4401}$

Order 1467: $C_{163}:C_9$ x 2, $C_{489}:C_3$, $C_{1467}$

Order 489: $C_{163}:C_3$ x 3, $C_{489}$

Order 243: $C_3\times C_{81}$

Order 163: $C_{163}$

Order 81: $C_{81}$ x 3, $C_3\times C_{27}$

Order 27: $C_{27}$ x 3, $C_3\times C_9$

Order 9: $C_9$ x 3, $C_3^2$

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 39609: $C_{4401}.C_9$

Order 13203: $C_{163}:C_{81}$, $C_{4401}:C_3$

Order 4401: $C_{163}:C_{27}$ x 2, $C_{1467}:C_3$, $C_{4401}$

Order 1467: $C_{163}:C_9$ x 2, $C_{489}:C_3$, $C_{1467}$

Order 489: $C_{489}$, $C_{163}:C_3$

Order 243: $C_3\times C_{81}$

Order 163: $C_{163}$

Order 81: $C_3\times C_{27}$, $C_{81}$

Order 27: $C_{27}$ x 3, $C_3\times C_9$

Order 9: $C_9$ x 3, $C_3^2$

Order 3: $C_3$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 39609: $C_{4401}.C_9$ ($C_1$)

Order 13203: $C_{163}:C_{81}$ ($C_3$) x 3, $C_{4401}:C_3$ ($C_3$)

Order 4401: $C_{1467}:C_3$ ($C_9$), $C_{163}:C_{27}$ ($C_3^2$), $C_{163}:C_{27}$ ($C_9$), $C_{4401}$ ($C_9$)

Order 1467: $C_{163}:C_9$ ($C_{27}$) x 2, $C_{489}:C_3$ ($C_{27}$), $C_{1467}$ ($C_3\times C_9$)

Order 489: $C_{163}:C_3$ ($C_{81}$) x 3, $C_{489}$ ($C_3\times C_{27}$)

Order 163: $C_{163}$ ($C_3\times C_{81}$)

Order 27: $C_{27}$ ($C_{163}:C_9$)

Order 9: $C_9$ ($C_{489}:C_9$)

Order 3: $C_3$ ($C_{1467}.C_9$)

Order 1: $C_1$ ($C_{4401}.C_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 39609: $C_{4401}.C_9$ ($C_1$)

Order 13203: $C_{163}:C_{81}$ ($C_3$), $C_{4401}:C_3$ ($C_3$)

Order 4401: $C_{1467}:C_3$ ($C_9$), $C_{163}:C_{27}$ ($C_3^2$), $C_{163}:C_{27}$ ($C_9$), $C_{4401}$ ($C_9$)

Order 1467: $C_{163}:C_9$ ($C_{27}$) x 2, $C_{489}:C_3$ ($C_{27}$), $C_{1467}$ ($C_3\times C_9$)

Order 489: $C_{489}$ ($C_3\times C_{27}$), $C_{163}:C_3$ ($C_{81}$)

Order 163: $C_{163}$ ($C_3\times C_{81}$)

Order 27: $C_{27}$ ($C_{163}:C_9$)

Order 9: $C_9$ ($C_{489}:C_9$)

Order 3: $C_3$ ($C_{1467}.C_9$)

Order 1: $C_1$ ($C_{4401}.C_9$)

Series

Derived series

$C_{4401}.C_9$

$\rhd$

$C_{163}$

$\rhd$

$C_1$

Chief series

$C_{4401}.C_9$

$\rhd$

$C_{163}:C_{81}$

$\rhd$

$C_{163}:C_{27}$

$\rhd$

$C_{1467}$

$\rhd$

$C_{489}$

$\rhd$

$C_{163}$

$\rhd$

$C_1$

Lower central series

$C_{4401}.C_9$

$\rhd$

$C_{163}$

Upper central series

$C_1$

$\lhd$

$C_{27}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $729 \times 729$ character table is not available for this group.

Rational character table

The $18 \times 18$ rational character table is not available for this group.