Abstract group 324.151: $C_3\times C_6\times C_{18}$

• Label: 324.151
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{7} \)
• Perm deg.: $19$
• Trans deg.: $324$
• Rank: $3$

Group information

Description:	$C_3\times C_6\times C_{18}$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^3.\ASL(2,3).C_6.C_2^2$, of order \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group).

Group statistics

Order	1	2	3	6	9	18
Elements	1	3	26	78	54	162	324
Conjugacy classes	1	3	26	78	54	162	324
Divisions	1	3	13	39	9	27	92
Autjugacy classes	1	1	2	2	1	1	8

Dimension	1	2	6
Irr. complex chars.	324	0	0	324
Irr. rational chars.	4	52	36	92

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$324$
Rank:	$3$
Inequivalent generating triples:	$91$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{3}=b^{6}=c^{18}=1 \rangle$
Permutation group:	Degree $19$ $\langle(1,2), (3,4), (11,19,16,13,18,15,12,17,14), (5,7,6), (8,10,9), (11,13,12)(14,16,15)(17,19,18)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 27 \\ 27 & 34 \end{array}\right), \left(\begin{array}{rr} 17 & 18 \\ 18 & 35 \end{array}\right), \left(\begin{array}{rr} 35 & 18 \\ 18 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ ${}^2$ $\, \times\, $ $C_9$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_3^2$	$C_3^2$ . $C_6^2$	$(C_3\times C_6^2)$ . $C_3$	$(C_3^2\times C_6)$ . $C_6$	all 9

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

Homology

Primary decomposition:	$C_{2}^{2} \times C_{3}^{2} \times C_{9}$
Schur multiplier:	$C_{3}^{2} \times C_{6}$
Commutator length:	$0$

Subgroups

There are 250 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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

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Classes of subgroups up to conjugation

Order 324: $C_3\times C_6\times C_{18}$

Order 162: $C_3^2\times C_{18}$ x 3

Order 108: $C_6\times C_{18}$ x 12, $C_3\times C_6^2$

Order 81: $C_3^2\times C_9$

Order 54: $C_3\times C_{18}$ x 36, $C_3^2\times C_6$ x 3

Order 36: $C_6^2$ x 13, $C_2\times C_{18}$ x 9

Order 27: $C_3\times C_9$ x 12, $C_3^3$

Order 18: $C_3\times C_6$ x 39, $C_{18}$ x 27

Order 12: $C_2\times C_6$ x 13

Order 9: $C_3^2$ x 13, $C_9$ x 9

Order 6: $C_6$ x 39

Order 4: $C_2^2$

Order 3: $C_3$ x 13

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324: $C_3\times C_6\times C_{18}$

Order 162: $C_3^2\times C_{18}$

Order 108: $C_3\times C_6^2$, $C_6\times C_{18}$

Order 81: $C_3^2\times C_9$

Order 54: $C_3\times C_{18}$, $C_3^2\times C_6$

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

Order 27: $C_3^3$, $C_3\times C_9$

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

Order 12: $C_2\times C_6$ x 2

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

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: $C_3\times C_6\times C_{18}$ ($C_1$)

Order 162: $C_3^2\times C_{18}$ ($C_2$) x 3

Order 108: $C_6\times C_{18}$ ($C_3$) x 12, $C_3\times C_6^2$ ($C_3$)

Order 81: $C_3^2\times C_9$ ($C_2^2$)

Order 54: $C_3\times C_{18}$ ($C_6$) x 36, $C_3^2\times C_6$ ($C_6$) x 3

Order 36: $C_2\times C_{18}$ ($C_3^2$) x 9, $C_6^2$ ($C_9$) x 9, $C_6^2$ ($C_3^2$) x 4

Order 27: $C_3\times C_9$ ($C_2\times C_6$) x 12, $C_3^3$ ($C_2\times C_6$)

Order 18: $C_3\times C_6$ ($C_{18}$) x 27, $C_{18}$ ($C_3\times C_6$) x 27, $C_3\times C_6$ ($C_3\times C_6$) x 12

Order 12: $C_2\times C_6$ ($C_3\times C_9$) x 12, $C_2\times C_6$ ($C_3^3$)

Order 9: $C_3^2$ ($C_2\times C_{18}$) x 9, $C_9$ ($C_6^2$) x 9, $C_3^2$ ($C_6^2$) x 4

Order 6: $C_6$ ($C_3\times C_{18}$) x 36, $C_6$ ($C_3^2\times C_6$) x 3

Order 4: $C_2^2$ ($C_3^2\times C_9$)

Order 3: $C_3$ ($C_6\times C_{18}$) x 12, $C_3$ ($C_3\times C_6^2$)

Order 2: $C_2$ ($C_3^2\times C_{18}$) x 3

Order 1: $C_1$ ($C_3\times C_6\times C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $C_3\times C_6\times C_{18}$ ($C_1$)

Order 162: $C_3^2\times C_{18}$ ($C_2$)

Order 108: $C_3\times C_6^2$ ($C_3$), $C_6\times C_{18}$ ($C_3$)

Order 81: $C_3^2\times C_9$ ($C_2^2$)

Order 54: $C_3\times C_{18}$ ($C_6$), $C_3^2\times C_6$ ($C_6$)

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

Order 27: $C_3^3$ ($C_2\times C_6$), $C_3\times C_9$ ($C_2\times C_6$)

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

Order 12: $C_2\times C_6$ ($C_3^3$), $C_2\times C_6$ ($C_3\times C_9$)

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

Order 6: $C_6$ ($C_3\times C_{18}$), $C_6$ ($C_3^2\times C_6$)

Order 4: $C_2^2$ ($C_3^2\times C_9$)

Order 3: $C_3$ ($C_3\times C_6^2$), $C_3$ ($C_6\times C_{18}$)

Order 2: $C_2$ ($C_3^2\times C_{18}$)

Order 1: $C_1$ ($C_3\times C_6\times C_{18}$)

Series

Derived series

$C_3\times C_6\times C_{18}$

$\rhd$

$C_1$

Chief series

$C_3\times C_6\times C_{18}$

$\rhd$

$C_6\times C_{18}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\times C_6\times C_{18}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6\times C_{18}$

Supergroups

This group is a maximal subgroup of 78 larger groups in the database.

This group is a maximal quotient of 19 larger groups in the database.

Character theory

Complex character table

See the $324 \times 324$ 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 $92 \times 92$ rational character table.