Abstract group 972.34: $C_{27}:C_{36}$

• Label: 972.34
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{3} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_{27}:C_{36}$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Automorphism group:	$C_{54}:C_{18}$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	9	12	18	27	36	54
Elements	1	1	8	54	8	72	108	72	162	324	162	972
Conjugacy classes	1	1	3	2	3	9	4	9	9	12	9	62
Divisions	1	1	2	1	2	3	1	3	3	1	3	21
Autjugacy classes	1	1	3	1	3	9	2	9	9	6	9	53

Dimension	1	2	4	6	12	18
Irr. complex chars.	36	18	0	6	0	2	62
Irr. rational chars.	2	5	3	4	5	2	21

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	18	20	20

Constructions

Presentation:	$\langle a, b \mid a^{36}=b^{27}=1, b^{a}=b^{2} \rangle$
Permutation group:	Degree $31$ $\langle(2,5)(3,11)(4,15)(6,8)(7,16)(9,21)(10,25)(12,13)(14,26)(17,22)(18,19)(20,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 80 & 95 \\ 27 & 82 \end{array}\right), \left(\begin{array}{rr} 1 & 54 \\ 54 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 73 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/108\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{27}$ $\,\rtimes\,$ $C_{36}$	$(C_{27}:C_9)$ $\,\rtimes\,$ $C_4$	$(C_{27}:C_4)$ $\,\rtimes\,$ $C_9$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{54}$ . $C_{18}$	$(C_9:C_{18})$ . $S_3$	$C_{18}$ . $(S_3\times C_9)$	$(C_{27}:C_6)$ . $C_6$	all 16

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

Homology

Abelianization:	$C_{36} \simeq C_{4} \times C_{9}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 192 subgroups in 44 conjugacy classes, 21 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{27}:C_{18}$
Commutator:	$G' \simeq$ $C_{27}$	$G/G' \simeq$ $C_{36}$
Frattini:	$\Phi \simeq$ $C_3\times C_{18}$	$G/\Phi \simeq$ $C_3\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{27}:C_{18}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{27}:C_{36}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_9:C_{18}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_{27}:C_9$

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 972: $C_{27}:C_{36}$

Order 486: $C_{27}:C_{18}$

Order 324: $C_9:C_{36}$, $C_{27}:C_{12}$

Order 243: $C_{27}:C_9$

Order 162: $C_{27}:C_6$ x 2, $C_9:C_{18}$

Order 108: $C_3:C_{36}$, $C_9:C_{12}$, $C_{27}:C_4$

Order 81: $C_{27}:C_3$ x 2, $C_9:C_9$

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

Order 36: $C_3:C_{12}$, $C_{36}$, $C_9:C_4$

Order 27: $C_{27}$ x 3, $C_3\times C_9$ x 2

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

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

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

Order 6: $C_6$ x 2

Order 4: $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $C_{27}:C_{36}$

Order 486: $C_{27}:C_{18}$

Order 324: $C_9:C_{36}$, $C_{27}:C_{12}$

Order 243: $C_{27}:C_9$

Order 162: $C_{27}:C_6$ x 2, $C_9:C_{18}$

Order 108: $C_3:C_{36}$, $C_9:C_{12}$, $C_{27}:C_4$

Order 81: $C_{27}:C_3$ x 2, $C_9:C_9$

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

Order 36: $C_3:C_{12}$, $C_{36}$, $C_9:C_4$

Order 27: $C_{27}$ x 3, $C_3\times C_9$ x 2

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

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

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

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 972: $C_{27}:C_{36}$ ($C_1$)

Order 486: $C_{27}:C_{18}$ ($C_2$)

Order 324: $C_{27}:C_{12}$ ($C_3$)

Order 243: $C_{27}:C_9$ ($C_4$)

Order 162: $C_{27}:C_6$ ($C_6$), $C_9:C_{18}$ ($S_3$)

Order 108: $C_{27}:C_4$ ($C_9$)

Order 81: $C_{27}:C_3$ ($C_{12}$), $C_9:C_9$ ($C_3:C_4$)

Order 54: $C_3\times C_{18}$ ($C_3\times S_3$), $C_{54}$ ($C_{18}$)

Order 27: $C_3\times C_9$ ($C_3:C_{12}$), $C_{27}$ ($C_{36}$)

Order 18: $C_3\times C_6$ ($C_9:C_6$), $C_{18}$ ($S_3\times C_9$)

Order 9: $C_3^2$ ($C_9:C_{12}$), $C_9$ ($C_3:C_{36}$)

Order 6: $C_6$ ($C_9:C_{18}$)

Order 3: $C_3$ ($C_9:C_{36}$)

Order 2: $C_2$ ($C_{27}:C_{18}$)

Order 1: $C_1$ ($C_{27}:C_{36}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $C_{27}:C_{36}$ ($C_1$)

Order 486: $C_{27}:C_{18}$ ($C_2$)

Order 324: $C_{27}:C_{12}$ ($C_3$)

Order 243: $C_{27}:C_9$ ($C_4$)

Order 162: $C_{27}:C_6$ ($C_6$), $C_9:C_{18}$ ($S_3$)

Order 108: $C_{27}:C_4$ ($C_9$)

Order 81: $C_{27}:C_3$ ($C_{12}$), $C_9:C_9$ ($C_3:C_4$)

Order 54: $C_3\times C_{18}$ ($C_3\times S_3$), $C_{54}$ ($C_{18}$)

Order 27: $C_3\times C_9$ ($C_3:C_{12}$), $C_{27}$ ($C_{36}$)

Order 18: $C_3\times C_6$ ($C_9:C_6$), $C_{18}$ ($S_3\times C_9$)

Order 9: $C_3^2$ ($C_9:C_{12}$), $C_9$ ($C_3:C_{36}$)

Order 6: $C_6$ ($C_9:C_{18}$)

Order 3: $C_3$ ($C_9:C_{36}$)

Order 2: $C_2$ ($C_{27}:C_{18}$)

Order 1: $C_1$ ($C_{27}:C_{36}$)

Series

Derived series

$C_{27}:C_{36}$

$\rhd$

$C_{27}$

$\rhd$

$C_1$

Chief series

$C_{27}:C_{36}$

$\rhd$

$C_{27}:C_{18}$

$\rhd$

$C_{27}:C_9$

$\rhd$

$C_{27}:C_3$

$\rhd$

$C_{27}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{27}:C_{36}$

$\rhd$

$C_{27}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $62 \times 62$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.