Abstract group 108.12: $C_3\times C_{36}$

• Label: 108.12
• Order: \( 2^{2} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{3} \)
• Perm deg.: $16$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_3\times C_{36}$
Order:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_6^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 3
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	1	8	2	8	18	16	18	36	108
Conjugacy classes	1	1	8	2	8	18	16	18	36	108
Divisions	1	1	4	1	4	3	4	3	3	24
Autjugacy classes	1	1	2	1	2	1	2	1	1	12

Dimension	1	2	4	6	12
Irr. complex chars.	108	0	0	0	0	108
Irr. rational chars.	2	9	4	6	3	24

Minimal presentations

Permutation degree:	$16$
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	2	4	10

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{3}=b^{36}=1 \rangle$
Permutation group:	Degree $16$ $\langle(1,4,2,3), (8,16,13,10,15,12,9,14,11), (5,7,6), (1,2)(3,4), (8,10,9)(11,13,12)(14,16,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 27 & 0 \\ 0 & 22 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{37})$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \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:	$C_{18}$ . $C_6$ (3)	$C_6$ . $C_{18}$ (3)	$C_{12}$ . $C_3^2$	$C_3^2$ . $C_{12}$	all 10

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

Homology

Primary decomposition:	$C_{4} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{3}$
Commutator length:	$0$

Subgroups

There are 30 subgroups, all normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_{36}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_3\times C_{36}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{36}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_3\times C_{36}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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 108: $C_3\times C_{36}$

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

Order 36: $C_{36}$ x 3, $C_3\times C_{12}$

Order 27: $C_3\times C_9$

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

Order 12: $C_{12}$ x 4

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

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 108: $C_3\times C_{36}$

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

Order 36: $C_3\times C_{12}$, $C_{36}$

Order 27: $C_3\times C_9$

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

Order 12: $C_{12}$ x 2

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

Order 6: $C_6$ x 2

Order 4: $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 108: $C_3\times C_{36}$ ($C_1$)

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

Order 36: $C_{36}$ ($C_3$) x 3, $C_3\times C_{12}$ ($C_3$)

Order 27: $C_3\times C_9$ ($C_4$)

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

Order 12: $C_{12}$ ($C_9$) x 3, $C_{12}$ ($C_3^2$)

Order 9: $C_9$ ($C_{12}$) x 3, $C_3^2$ ($C_{12}$)

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

Order 4: $C_4$ ($C_3\times C_9$)

Order 3: $C_3$ ($C_{36}$) x 3, $C_3$ ($C_3\times C_{12}$)

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

Order 1: $C_1$ ($C_3\times C_{36}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 108: $C_3\times C_{36}$ ($C_1$)

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

Order 36: $C_3\times C_{12}$ ($C_3$), $C_{36}$ ($C_3$)

Order 27: $C_3\times C_9$ ($C_4$)

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

Order 12: $C_{12}$ ($C_3^2$), $C_{12}$ ($C_9$)

Order 9: $C_3^2$ ($C_{12}$), $C_9$ ($C_{12}$)

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

Order 4: $C_4$ ($C_3\times C_9$)

Order 3: $C_3$ ($C_3\times C_{12}$), $C_3$ ($C_{36}$)

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

Order 1: $C_1$ ($C_3\times C_{36}$)

Series

Derived series

$C_3\times C_{36}$

$\rhd$

$C_1$

Chief series

$C_3\times C_{36}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3\times C_{36}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3\times C_{36}$

Supergroups

This group is a maximal subgroup of 106 larger groups in the database.

This group is a maximal quotient of 83 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.