Abstract group 324.85: $C_{54}:C_6$

• Label: 324.85
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{3} \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_{54}:C_6$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(54\)\(\medspace = 2 \cdot 3^{3} \)
Automorphism group:	$(C_3\times C_9):S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metacyclic (hence metabelian).

Group statistics

Order	1	2	3	6	9	18	27	54
Elements	1	3	8	24	18	54	54	162	324
Conjugacy classes	1	3	4	12	10	30	18	54	132
Divisions	1	3	2	6	3	9	3	9	36
Autjugacy classes	1	1	3	3	3	3	1	1	16

Dimension	1	2	3	6	18
Irr. complex chars.	108	0	24	0	0	132
Irr. rational chars.	4	16	0	12	4	36

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	7	19

Constructions

Presentation:	$\langle a, b \mid a^{6}=b^{54}=1, b^{a}=b^{19} \rangle$
Permutation group:	Degree $31$ $\langle(1,2), (3,4), (5,31,22,13,28,19,10,25,16,7,30,21,12,27,18,9,24,15,6,29,20,11,26,17,8,23,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 53 & 0 \\ 0 & 53 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_{27}:C_3)$
Semidirect product:	$C_{54}$ $\,\rtimes\,$ $C_6$ (9)	$(C_2\times C_{54})$ $\,\rtimes\,$ $C_3$ (3)	$C_{27}$ $\,\rtimes\,$ $(C_2\times C_6)$ (3)		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_9$	$C_{18}$ . $C_{18}$ (6)	$C_9$ . $C_6^2$	$(C_6\times C_{18})$ . $C_3$	all 15

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 70 subgroups in 60 conjugacy classes, 55 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{18}$	$G/Z \simeq$ $C_3^2$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_9$	$G/\Phi \simeq$ $C_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{54}:C_6$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{54}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times C_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_{27}:C_3$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 324: $C_{54}:C_6$

Order 162: $C_{27}:C_6$ x 3

Order 108: $C_2\times C_{54}$ x 3, $C_6\times C_{18}$

Order 81: $C_{27}:C_3$

Order 54: $C_{54}$ x 9, $C_3\times C_{18}$ x 3

Order 36: $C_2\times C_{18}$ x 3, $C_6^2$

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

Order 18: $C_{18}$ x 9, $C_3\times C_6$ x 3

Order 12: $C_2\times C_6$ x 2

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

Order 6: $C_6$ x 6

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324: $C_{54}:C_6$

Order 162: $C_{27}:C_6$

Order 108: $C_2\times C_{54}$, $C_6\times C_{18}$

Order 81: $C_{27}:C_3$

Order 54: $C_3\times C_{18}$, $C_{54}$

Order 36: $C_2\times C_{18}$ x 2, $C_6^2$

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

Order 18: $C_{18}$ x 2, $C_3\times C_6$

Order 12: $C_2\times C_6$ x 2

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

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 324: $C_{54}:C_6$ ($C_1$)

Order 162: $C_{27}:C_6$ ($C_2$) x 3

Order 108: $C_2\times C_{54}$ ($C_3$) x 3, $C_6\times C_{18}$ ($C_3$)

Order 81: $C_{27}:C_3$ ($C_2^2$)

Order 54: $C_{54}$ ($C_6$) x 9, $C_3\times C_{18}$ ($C_6$) x 3

Order 36: $C_2\times C_{18}$ ($C_9$) x 2, $C_2\times C_{18}$ ($C_3^2$), $C_6^2$ ($C_9$)

Order 27: $C_{27}$ ($C_2\times C_6$) x 3, $C_3\times C_9$ ($C_2\times C_6$)

Order 18: $C_{18}$ ($C_{18}$) x 6, $C_3\times C_6$ ($C_{18}$) x 3, $C_{18}$ ($C_3\times C_6$) x 3

Order 12: $C_2\times C_6$ ($C_3\times C_9$)

Order 9: $C_9$ ($C_2\times C_{18}$) x 2, $C_3^2$ ($C_2\times C_{18}$), $C_9$ ($C_6^2$)

Order 6: $C_6$ ($C_3\times C_{18}$) x 3

Order 4: $C_2^2$ ($C_{27}:C_3$)

Order 3: $C_3$ ($C_6\times C_{18}$)

Order 2: $C_2$ ($C_{27}:C_6$) x 3

Order 1: $C_1$ ($C_{54}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $C_{54}:C_6$ ($C_1$)

Order 162: $C_{27}:C_6$ ($C_2$)

Order 108: $C_2\times C_{54}$ ($C_3$), $C_6\times C_{18}$ ($C_3$)

Order 81: $C_{27}:C_3$ ($C_2^2$)

Order 54: $C_3\times C_{18}$ ($C_6$), $C_{54}$ ($C_6$)

Order 36: $C_2\times C_{18}$ ($C_3^2$), $C_2\times C_{18}$ ($C_9$), $C_6^2$ ($C_9$)

Order 27: $C_3\times C_9$ ($C_2\times C_6$), $C_{27}$ ($C_2\times C_6$)

Order 18: $C_3\times C_6$ ($C_{18}$), $C_{18}$ ($C_3\times C_6$), $C_{18}$ ($C_{18}$)

Order 12: $C_2\times C_6$ ($C_3\times C_9$)

Order 9: $C_3^2$ ($C_2\times C_{18}$), $C_9$ ($C_2\times C_{18}$), $C_9$ ($C_6^2$)

Order 6: $C_6$ ($C_3\times C_{18}$)

Order 4: $C_2^2$ ($C_{27}:C_3$)

Order 3: $C_3$ ($C_6\times C_{18}$)

Order 2: $C_2$ ($C_{27}:C_6$)

Order 1: $C_1$ ($C_{54}:C_6$)

Series

Derived series

$C_{54}:C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{54}:C_6$

$\rhd$

$C_{27}:C_6$

$\rhd$

$C_{54}$

$\rhd$

$C_{27}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{54}:C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{18}$

$\lhd$

$C_{54}:C_6$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $132 \times 132$ 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 $36 \times 36$ rational character table.