Abstract group 486.254: $C_6.C_3^4$

• Label: 486.254
• Order: \( 2 \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{4} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• Perm deg.: $29$
• Trans deg.: $54$
• Rank: $4$

Group information

Description:	$C_6.C_3^4$
Order:	\(486\)\(\medspace = 2 \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$, $C_3$ x 5
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	6
Elements	1	1	242	242	486
Conjugacy classes	1	1	82	82	166
Divisions	1	1	41	41	84
Autjugacy classes	1	1	2	2	6

Dimension	1	2	9	18
Irr. complex chars.	162	0	4	0	166
Irr. rational chars.	2	80	0	2	84

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$54$
Rank:	$4$
Inequivalent generating quadruples:	$3510$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{3}=b^{3}=c^{6}=d^{3}=e^{3}=[a,b]=[a,d]=[a,e]=[b,c]=[b,e]=[c,d]=[c,e]=[d,e]=1, c^{a}=ce^{2}, d^{b}=de^{2} \rangle$
Permutation group:	Degree $29$ $\langle(1,2), (3,15,29)(4,16,27)(5,17,28)(6,19,22)(7,20,23)(8,18,21)(9,14,24)(10,12,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 2 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 2 \\ 1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 2 & 0 \\ 0 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Direct product:	$C_2$ $\, \times\, $ $(C_3^3:C_3^2)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $(C_3\times C_6)$	$(C_6\times \He_3)$ $\,\rtimes\,$ $C_3$	$(C_3\times \He_3)$ $\,\rtimes\,$ $C_6$	$\He_3$ $\,\rtimes\,$ $(C_3\times C_6)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $C_3^4$	$(C_3\times C_6)$ . $C_3^3$	$C_3$ . $(C_3^3\times C_6)$	$C_3^2$ . $(C_3^2\times C_6)$	more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

Abelianization:	$C_{3}^{3} \times C_{6} \simeq C_{2} \times C_{3}^{4}$
Schur multiplier:	$C_{3}^{5}$
Commutator length:	$1$

Subgroups

There are 1386 subgroups in 586 conjugacy classes, 426 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^4$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_3^3\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_6.C_3^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3:C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

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.

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

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

Classes of subgroups up to conjugation

Order 486: $C_6.C_3^4$

Order 243: $C_3^3:C_3^2$

Order 162: $C_6\times \He_3$ x 40

Order 81: $C_3\times \He_3$ x 40

Order 54: $C_2\times \He_3$ x 90, $C_3^2\times C_6$ x 40

Order 27: $\He_3$ x 90, $C_3^3$ x 40

Order 18: $C_3\times C_6$ x 80

Order 9: $C_3^2$ x 80

Order 6: $C_6$ x 41

Order 3: $C_3$ x 41

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 486: $C_6.C_3^4$

Order 243: $C_3^3:C_3^2$

Order 162: $C_6\times \He_3$

Order 81: $C_3\times \He_3$

Order 54: $C_3^2\times C_6$, $C_2\times \He_3$

Order 27: $C_3^3$, $\He_3$

Order 18: $C_3\times C_6$ x 2

Order 9: $C_3^2$ x 2

Order 6: $C_6$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 486: $C_6.C_3^4$ ($C_1$)

Order 243: $C_3^3:C_3^2$ ($C_2$)

Order 162: $C_6\times \He_3$ ($C_3$) x 40

Order 81: $C_3\times \He_3$ ($C_6$) x 40

Order 54: $C_2\times \He_3$ ($C_3^2$) x 90, $C_3^2\times C_6$ ($C_3^2$) x 40

Order 27: $\He_3$ ($C_3\times C_6$) x 90, $C_3^3$ ($C_3\times C_6$) x 40

Order 18: $C_3\times C_6$ ($C_3^3$) x 40

Order 9: $C_3^2$ ($C_3^2\times C_6$) x 40

Order 6: $C_6$ ($C_3^4$)

Order 3: $C_3$ ($C_3^3\times C_6$)

Order 2: $C_2$ ($C_3^3:C_3^2$)

Order 1: $C_1$ ($C_6.C_3^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 486: $C_6.C_3^4$ ($C_1$)

Order 243: $C_3^3:C_3^2$ ($C_2$)

Order 162: $C_6\times \He_3$ ($C_3$)

Order 81: $C_3\times \He_3$ ($C_6$)

Order 54: $C_3^2\times C_6$ ($C_3^2$), $C_2\times \He_3$ ($C_3^2$)

Order 27: $C_3^3$ ($C_3\times C_6$), $\He_3$ ($C_3\times C_6$)

Order 18: $C_3\times C_6$ ($C_3^3$)

Order 9: $C_3^2$ ($C_3^2\times C_6$)

Order 6: $C_6$ ($C_3^4$)

Order 3: $C_3$ ($C_3^3\times C_6$)

Order 2: $C_2$ ($C_3^3:C_3^2$)

Order 1: $C_1$ ($C_6.C_3^4$)

Series

Derived series

$C_6.C_3^4$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_6.C_3^4$

$\rhd$

$C_3^3:C_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.C_3^4$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_6.C_3^4$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $166 \times 166$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $84 \times 84$ rational character table.