Abstract group 432.131: $C_{12}:C_{36}$

• Label: 432.131
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$C_{12}:C_{36}$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_6^2.C_2^5$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
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	9	12	18	36
Elements	1	3	8	28	24	18	80	54	216	432
Conjugacy classes	1	3	5	12	15	12	36	36	96	216
Divisions	1	3	3	6	9	2	10	6	8	48
Autjugacy classes	1	2	3	2	6	2	4	4	3	27

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	144	72	0	0	0	0	0	216
Irr. rational chars.	4	14	12	4	2	10	2	48

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	12

Constructions

Presentation:	$\langle a, b \mid a^{36}=b^{12}=1, b^{a}=b^{5} \rangle$
Permutation group:	Degree $20$ $\langle(10,11,12,13)(15,16)(17,18)(19,20), (17,19,18,20), (1,2,4,3,5,7,6,8,9), (10,12)(11,13), (17,18)(19,20), (1,3,6)(2,5,8)(4,7,9), (14,15,16)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 20 & 19 \\ 19 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 20 & 0 \\ 0 & 20 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 30 \\ 12 & 19 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/57\Z)$
Direct product:	$C_4$ $\, \times\, $ $C_9$ $\, \times\, $ $(C_3:C_4)$
Semidirect product:	$C_{12}$ $\,\rtimes\,$ $C_{36}$ (2)	$(C_3\times C_{36})$ $\,\rtimes\,$ $C_4$ (2)	$C_3$ $\,\rtimes\,$ $(C_4\times C_{36})$	$(C_3\times C_9)$ $\,\rtimes\,$ $C_4^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{36})$ . $S_3$	$C_{18}$ . $(C_4\times S_3)$ (2)	$C_6$ . $(S_3\times C_{12})$ (2)	$C_2$ . $(S_3\times C_{36})$ (2)	all 30

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

Homology

Abelianization:	$C_{4} \times C_{36} \simeq C_{4}^{2} \times C_{9}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 164 subgroups in 106 conjugacy classes, 69 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{36}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_4\times C_{36}$
Frattini:	$\Phi \simeq$ $C_2\times C_6$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{36}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}:C_{36}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 432: $C_{12}:C_{36}$

Order 216: $C_6:C_{36}$ x 2, $C_6\times C_{36}$

Order 144: $C_{12}:C_{12}$, $C_4\times C_{36}$

Order 108: $C_3:C_{36}$ x 4, $C_3\times C_{36}$ x 2, $C_6\times C_{18}$

Order 72: $C_2\times C_{36}$ x 4, $C_6:C_{12}$ x 2, $C_6\times C_{12}$

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

Order 48: $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $C_{36}$ x 8, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2, $C_2\times C_{18}$ x 2, $C_6^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 5, $C_6:C_4$ x 2

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

Order 16: $C_4^2$

Order 12: $C_{12}$ x 10, $C_3:C_4$ x 4, $C_2\times C_6$ x 3

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

Order 8: $C_2\times C_4$ x 3

Order 6: $C_6$ x 9

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $C_{12}:C_{36}$

Order 216: $C_6\times C_{36}$, $C_6:C_{36}$

Order 144: $C_{12}:C_{12}$, $C_4\times C_{36}$

Order 108: $C_3:C_{36}$, $C_6\times C_{18}$, $C_3\times C_{36}$

Order 72: $C_2\times C_{36}$ x 3, $C_6\times C_{12}$, $C_6:C_{12}$

Order 54: $C_3\times C_{18}$ x 2

Order 48: $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $C_{36}$ x 3, $C_2\times C_{18}$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $C_6^2$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 4, $C_6:C_4$

Order 18: $C_{18}$ x 4, $C_3\times C_6$ x 2

Order 16: $C_4^2$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3, $C_3:C_4$

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

Order 8: $C_2\times C_4$ x 2

Order 6: $C_6$ x 6

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_{12}:C_{36}$ ($C_1$)

Order 216: $C_6:C_{36}$ ($C_2$) x 2, $C_6\times C_{36}$ ($C_2$)

Order 144: $C_{12}:C_{12}$ ($C_3$)

Order 108: $C_3:C_{36}$ ($C_4$) x 4, $C_3\times C_{36}$ ($C_4$) x 2, $C_6\times C_{18}$ ($C_2^2$)

Order 72: $C_6:C_{12}$ ($C_6$) x 2, $C_2\times C_{36}$ ($S_3$), $C_6\times C_{12}$ ($C_6$)

Order 54: $C_3\times C_{18}$ ($C_2\times C_4$) x 3

Order 48: $C_{12}:C_4$ ($C_9$)

Order 36: $C_3:C_{12}$ ($C_{12}$) x 4, $C_3\times C_{12}$ ($C_{12}$) x 2, $C_{36}$ ($C_3:C_4$) x 2, $C_2\times C_{18}$ ($D_6$), $C_6^2$ ($C_2\times C_6$)

Order 27: $C_3\times C_9$ ($C_4^2$)

Order 24: $C_6:C_4$ ($C_{18}$) x 2, $C_2\times C_{12}$ ($C_3\times S_3$), $C_2\times C_{12}$ ($C_{18}$)

Order 18: $C_3\times C_6$ ($C_2\times C_{12}$) x 3, $C_{18}$ ($C_4\times S_3$) x 2, $C_{18}$ ($C_6:C_4$)

Order 12: $C_3:C_4$ ($C_{36}$) x 4, $C_{12}$ ($C_3:C_{12}$) x 2, $C_{12}$ ($C_{36}$) x 2, $C_2\times C_6$ ($C_2\times C_{18}$), $C_2\times C_6$ ($C_6\times S_3$)

Order 9: $C_3^2$ ($C_4\times C_{12}$), $C_9$ ($C_{12}:C_4$)

Order 8: $C_2\times C_4$ ($S_3\times C_9$)

Order 6: $C_6$ ($C_2\times C_{36}$) x 3, $C_6$ ($S_3\times C_{12}$) x 2, $C_6$ ($C_6:C_{12}$)

Order 4: $C_4$ ($C_3:C_{36}$) x 2, $C_2^2$ ($S_3\times C_{18}$)

Order 3: $C_3$ ($C_{12}:C_{12}$), $C_3$ ($C_4\times C_{36}$)

Order 2: $C_2$ ($S_3\times C_{36}$) x 2, $C_2$ ($C_6:C_{36}$)

Order 1: $C_1$ ($C_{12}:C_{36}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_{12}:C_{36}$ ($C_1$)

Order 216: $C_6\times C_{36}$ ($C_2$), $C_6:C_{36}$ ($C_2$)

Order 144: $C_{12}:C_{12}$ ($C_3$)

Order 108: $C_3:C_{36}$ ($C_4$), $C_6\times C_{18}$ ($C_2^2$), $C_3\times C_{36}$ ($C_4$)

Order 72: $C_2\times C_{36}$ ($S_3$), $C_6\times C_{12}$ ($C_6$), $C_6:C_{12}$ ($C_6$)

Order 54: $C_3\times C_{18}$ ($C_2\times C_4$) x 2

Order 48: $C_{12}:C_4$ ($C_9$)

Order 36: $C_3\times C_{12}$ ($C_{12}$), $C_3:C_{12}$ ($C_{12}$), $C_2\times C_{18}$ ($D_6$), $C_{36}$ ($C_3:C_4$), $C_6^2$ ($C_2\times C_6$)

Order 27: $C_3\times C_9$ ($C_4^2$)

Order 24: $C_2\times C_{12}$ ($C_3\times S_3$), $C_2\times C_{12}$ ($C_{18}$), $C_6:C_4$ ($C_{18}$)

Order 18: $C_3\times C_6$ ($C_2\times C_{12}$) x 2, $C_{18}$ ($C_6:C_4$), $C_{18}$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($C_2\times C_{18}$), $C_2\times C_6$ ($C_6\times S_3$), $C_{12}$ ($C_3:C_{12}$), $C_{12}$ ($C_{36}$), $C_3:C_4$ ($C_{36}$)

Order 9: $C_3^2$ ($C_4\times C_{12}$), $C_9$ ($C_{12}:C_4$)

Order 8: $C_2\times C_4$ ($S_3\times C_9$)

Order 6: $C_6$ ($C_2\times C_{36}$) x 2, $C_6$ ($C_6:C_{12}$), $C_6$ ($S_3\times C_{12}$)

Order 4: $C_2^2$ ($S_3\times C_{18}$), $C_4$ ($C_3:C_{36}$)

Order 3: $C_3$ ($C_{12}:C_{12}$), $C_3$ ($C_4\times C_{36}$)

Order 2: $C_2$ ($C_6:C_{36}$), $C_2$ ($S_3\times C_{36}$)

Order 1: $C_1$ ($C_{12}:C_{36}$)

Series

Derived series

$C_{12}:C_{36}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{12}:C_{36}$

$\rhd$

$C_6\times C_{36}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}:C_{36}$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{36}$

Supergroups

This group is a maximal subgroup of 74 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

See the $216 \times 216$ 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 $48 \times 48$ rational character table.