Abstract group 396.13: $C_{33}:C_{12}$

• Label: 396.13
• Order: \( 2^{2} \cdot 3^{2} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $21$
• Trans deg.: $132$
• Rank: $2$

Group information

Description:	$C_{33}:C_{12}$
Order:	\(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \)
Exponent:	\(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \)
Automorphism group:	$C_{11}:(C_2^2\times C_{10}\times S_3)$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 2, $C_3$ x 2, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	11	12	22	33	66
Elements	1	1	8	66	8	10	132	10	80	80	396
Conjugacy classes	1	1	5	2	5	5	4	5	40	40	108
Divisions	1	1	3	1	3	1	1	1	3	3	18
Autjugacy classes	1	1	3	1	3	1	1	1	3	3	18

Dimension	1	2	4	10	20	40
Irr. complex chars.	12	96	0	0	0	0	108
Irr. rational chars.	2	5	3	2	4	2	18

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$132$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	40
Arbitrary	2	4	14

Constructions

Presentation:	$\langle a, b \mid a^{12}=b^{33}=1, b^{a}=b^{32} \rangle$
Permutation group:	Degree $21$ $\langle(2,3)(4,5,6,7)(12,13)(14,15)(16,17)(18,19)(20,21), (8,9,10), (4,6)(5,7), (1,2,3), (11,12,14,16,18,20,21,19,17,15,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 38 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 34 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{67})$
Direct product:	$C_3$ $\, \times\, $ $(C_{33}:C_4)$
Semidirect product:	$C_{33}$ $\,\rtimes\,$ $C_{12}$	$(C_3\times C_{33})$ $\,\rtimes\,$ $C_4$	$C_{33}$ $\,\rtimes\,$ $(C_3:C_4)$	$C_{11}$ $\,\rtimes\,$ $(C_3:C_{12})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{66}$ . $S_3$	$C_6$ . $D_{33}$	$C_{66}$ . $C_6$	$C_{22}$ . $(C_3\times S_3)$	all 8

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

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 120 subgroups in 28 conjugacy classes, 18 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_{33}$
Commutator:	$G' \simeq$ $C_{33}$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3\times D_{33}$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{66}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{33}:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{66}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 396: $C_{33}:C_{12}$

Order 198: $C_3\times C_{66}$

Order 132: $C_{33}:C_4$, $C_{11}:C_{12}$

Order 99: $C_3\times C_{33}$

Order 66: $C_{66}$ x 3

Order 44: $C_{11}:C_4$

Order 36: $C_3:C_{12}$

Order 33: $C_{33}$ x 3

Order 22: $C_{22}$

Order 18: $C_3\times C_6$

Order 12: $C_{12}$, $C_3:C_4$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 6: $C_6$ x 3

Order 4: $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 396: $C_{33}:C_{12}$

Order 198: $C_3\times C_{66}$

Order 132: $C_{33}:C_4$, $C_{11}:C_{12}$

Order 99: $C_3\times C_{33}$

Order 66: $C_{66}$ x 3

Order 44: $C_{11}:C_4$

Order 36: $C_3:C_{12}$

Order 33: $C_{33}$ x 3

Order 22: $C_{22}$

Order 18: $C_3\times C_6$

Order 12: $C_{12}$, $C_3:C_4$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 6: $C_6$ x 3

Order 4: $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 396: $C_{33}:C_{12}$ ($C_1$)

Order 198: $C_3\times C_{66}$ ($C_2$)

Order 132: $C_{33}:C_4$ ($C_3$)

Order 99: $C_3\times C_{33}$ ($C_4$)

Order 66: $C_{66}$ ($C_6$), $C_{66}$ ($S_3$)

Order 33: $C_{33}$ ($C_{12}$), $C_{33}$ ($C_3:C_4$)

Order 22: $C_{22}$ ($C_3\times S_3$)

Order 18: $C_3\times C_6$ ($D_{11}$)

Order 11: $C_{11}$ ($C_3:C_{12}$)

Order 9: $C_3^2$ ($C_{11}:C_4$)

Order 6: $C_6$ ($D_{33}$), $C_6$ ($C_3\times D_{11}$)

Order 3: $C_3$ ($C_{33}:C_4$), $C_3$ ($C_{11}:C_{12}$)

Order 2: $C_2$ ($C_3\times D_{33}$)

Order 1: $C_1$ ($C_{33}:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 396: $C_{33}:C_{12}$ ($C_1$)

Order 198: $C_3\times C_{66}$ ($C_2$)

Order 132: $C_{33}:C_4$ ($C_3$)

Order 99: $C_3\times C_{33}$ ($C_4$)

Order 66: $C_{66}$ ($C_6$), $C_{66}$ ($S_3$)

Order 33: $C_{33}$ ($C_{12}$), $C_{33}$ ($C_3:C_4$)

Order 22: $C_{22}$ ($C_3\times S_3$)

Order 18: $C_3\times C_6$ ($D_{11}$)

Order 11: $C_{11}$ ($C_3:C_{12}$)

Order 9: $C_3^2$ ($C_{11}:C_4$)

Order 6: $C_6$ ($D_{33}$), $C_6$ ($C_3\times D_{11}$)

Order 3: $C_3$ ($C_{33}:C_4$), $C_3$ ($C_{11}:C_{12}$)

Order 2: $C_2$ ($C_3\times D_{33}$)

Order 1: $C_1$ ($C_{33}:C_{12}$)

Series

Derived series

$C_{33}:C_{12}$

$\rhd$

$C_{33}$

$\rhd$

$C_1$

Chief series

$C_{33}:C_{12}$

$\rhd$

$C_3\times C_{66}$

$\rhd$

$C_3\times C_{33}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{33}:C_{12}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 35 larger groups in the database.

This group is a maximal quotient of 16 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 $18 \times 18$ rational character table.