Abstract group 333.3: $C_{37}:C_9$

• Label: 333.3
• Order: \( 3^{2} \cdot 37 \)
• Exponent: \( 3^{2} \cdot 37 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 37 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $37$
• Trans deg.: $37$
• Rank: $2$

Group information

Description:	$C_{37}:C_9$
Order:	\(333\)\(\medspace = 3^{2} \cdot 37 \)
Exponent:	\(333\)\(\medspace = 3^{2} \cdot 37 \)
Automorphism group:	$F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \)
Composition factors:	$C_3$ x 2, $C_{37}$
Derived length:	$2$

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

Group statistics

Order	1	3	9	37
Elements	1	74	222	36	333
Conjugacy classes	1	2	6	4	13
Divisions	1	1	1	1	4
Autjugacy classes	1	2	6	1	10

Dimension	1	2	6	9	36
Irr. complex chars.	9	0	0	4	0	13
Irr. rational chars.	1	1	1	0	1	4

Minimal presentations

Permutation degree:	$37$
Transitive degree:	$37$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	36
Arbitrary	9	18	36

Constructions

Presentation:	$\langle a, b \mid a^{9}=b^{37}=1, b^{a}=b^{9} \rangle$
Permutation group:	Degree $37$ $\langle(2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 12 & 0 \\ 0 & 26 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{37})$
Transitive group:	37T6				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{37}$ $\,\rtimes\,$ $C_9$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{37}:C_3)$ . $C_3$				more information

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

Homology

Abelianization:	$C_{9} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 78 subgroups in 6 conjugacy classes, 4 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{37}:C_9$
Commutator:	$G' \simeq$ $C_{37}$	$G/G' \simeq$ $C_9$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{37}:C_9$
Fitting:	$\operatorname{Fit} \simeq$ $C_{37}$	$G/\operatorname{Fit} \simeq$ $C_9$
Radical:	$R \simeq$ $C_{37}:C_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{37}$	$G/\operatorname{soc} \simeq$ $C_9$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

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 333: $C_{37}:C_9$

Order 111: $C_{37}:C_3$

Order 37: $C_{37}$

Order 9: $C_9$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 333: $C_{37}:C_9$

Order 111: $C_{37}:C_3$

Order 37: $C_{37}$

Order 9: $C_9$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 333: $C_{37}:C_9$ ($C_1$)

Order 111: $C_{37}:C_3$ ($C_3$)

Order 37: $C_{37}$ ($C_9$)

Order 1: $C_1$ ($C_{37}:C_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 333: $C_{37}:C_9$ ($C_1$)

Order 111: $C_{37}:C_3$ ($C_3$)

Order 37: $C_{37}$ ($C_9$)

Order 1: $C_1$ ($C_{37}:C_9$)

Series

Derived series	$C_{37}:C_9$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Chief series	$C_{37}:C_9$	$\rhd$	$C_{37}:C_3$	$\rhd$	$C_{37}$	$\rhd$	$C_1$
Lower central series	$C_{37}:C_9$	$\rhd$	$C_{37}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

		1A	3A1	3A-1	9A1	9A-1	9A2	9A-2	9A4	9A-4	37A1	37A-1	37A2	37A-2
	Size	1	37	37	37	37	37	37	37	37	9	9	9	9
	3 P	1A	3A-1	3A1	9A2	9A-2	9A4	9A-4	9A-1	9A1	37A2	37A-2	37A-1	37A1
	37 P	1A	1A	1A	3A1	3A-1	3A-1	3A1	3A1	3A-1	37A-1	37A1	37A-2	37A2
	Type
333.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
333.3.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3$	$1$	$1$	$1$	$1$
333.3.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$
333.3.1c1	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^{-1}$	${\zeta }_9^{-4}$	${\zeta }_9^2$	$1$	$1$	$1$	$1$
333.3.1c2	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9$	${\zeta }_9^4$	${\zeta }_9^{-2}$	$1$	$1$	$1$	$1$
333.3.1c3	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9$	${\zeta }_9^{-4}$	${\zeta }_9^2$	${\zeta }_9^{-1}$	$1$	$1$	$1$	$1$
333.3.1c4	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^{-1}$	${\zeta }_9^4$	${\zeta }_9^{-2}$	${\zeta }_9$	$1$	$1$	$1$	$1$
333.3.1c5	C	$1$	${\zeta }_9^{-3}$	${\zeta }_9^3$	${\zeta }_9$	${\zeta }_9^{-2}$	${\zeta }_9^4$	${\zeta }_9^2$	${\zeta }_9^{-1}$	${\zeta }_9^{-4}$	$1$	$1$	$1$	$1$
333.3.1c6	C	$1$	${\zeta }_9^3$	${\zeta }_9^{-3}$	${\zeta }_9^{-1}$	${\zeta }_9^2$	${\zeta }_9^{-4}$	${\zeta }_9^{-2}$	${\zeta }_9$	${\zeta }_9^4$	$1$	$1$	$1$	$1$
333.3.9a1	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$
333.3.9a2	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$
333.3.9a3	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$
333.3.9a4	C	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{37}^{-11}+{\zeta }_{37}^{-4}+{\zeta }_{37}^{-3}+{\zeta }_{37}+{\zeta }_{37}^7+{\zeta }_{37}^9+{\zeta }_{37}^{10}+{\zeta }_{37}^{12}+{\zeta }_{37}^{16}$	${\zeta }_{37}^{-16}+{\zeta }_{37}^{-12}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{-9}+{\zeta }_{37}^{-7}+{\zeta }_{37}^{-1}+{\zeta }_{37}^3+{\zeta }_{37}^4+{\zeta }_{37}^{11}$	${\zeta }_{37}^{-17}+{\zeta }_{37}^{-13}+{\zeta }_{37}^{-8}+{\zeta }_{37}^{-6}+{\zeta }_{37}^{-5}+{\zeta }_{37}^2+{\zeta }_{37}^{14}+{\zeta }_{37}^{15}+{\zeta }_{37}^{18}$	${\zeta }_{37}^{-18}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{-14}+{\zeta }_{37}^{-2}+{\zeta }_{37}^5+{\zeta }_{37}^6+{\zeta }_{37}^8+{\zeta }_{37}^{13}+{\zeta }_{37}^{17}$

Rational character table

		1A	3A	9A	37A
	Size	1	74	222	36
	3 P	1A	3A	9A	37A
	37 P	1A	1A	3A	37A
333.3.1a		$1$	$1$	$1$	$1$
333.3.1b		$2$	$2$	$-1$	$2$
333.3.1c		$6$	$-3$	$0$	$6$
333.3.9a		$36$	$0$	$0$	$-1$