Abstract group 648.114: $C_{36}.C_{18}$

• Label: 648.114
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$C_{36}.C_{18}$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_{36}:C_6^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
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	36
Elements	1	1	8	38	8	72	88	72	360	648
Conjugacy classes	1	1	5	3	5	15	12	15	36	93
Divisions	1	1	3	3	3	4	5	4	6	30
Autjugacy classes	1	1	5	2	5	7	7	7	9	44

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	36	45	0	12	0	0	0	93
Irr. rational chars.	4	7	4	6	1	6	2	30

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	24
Arbitrary	6	10	16

Constructions

Presentation:	$\langle a, b \mid b^{36}=1, a^{18}=b^{18}, b^{a}=b^{11} \rangle$
Permutation group:	Degree $26$ $\langle(10,11,13,15)(12,16,17,14)(19,21)(20,25)(22,23)(24,26), (10,12,13,17)(11,14,15,16) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_9:C_9)$ $\,\rtimes\,$ $Q_8$	$(C_9:Q_8)$ $\,\rtimes\,$ $C_9$	$C_9$ $\,\rtimes\,$ $(Q_8\times C_9)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{36}$ . $C_{18}$	$(C_3\times C_{36})$ . $S_3$	$C_{12}$ . $(S_3\times C_9)$	$C_6$ . $(S_3\times C_{18})$	all 25

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

Homology

Abelianization:	$C_{2} \times C_{18} \simeq C_{2}^{2} \times C_{9}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 186 subgroups in 66 conjugacy classes, 33 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_{18}:C_6$
Commutator:	$G' \simeq$ $C_{18}$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_9:C_{36}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{36}.C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_6\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_9$

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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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 648: $C_{36}.C_{18}$

Order 324: $C_9:C_{36}$ x 2, $C_9:C_{36}$

Order 216: $C_{12}.C_{18}$, $C_{36}.C_6$

Order 162: $C_9:C_{18}$

Order 108: $C_3\times C_{36}$ x 3, $C_3:C_{36}$ x 2, $C_9:C_{12}$ x 2

Order 81: $C_9:C_9$

Order 72: $C_9:Q_8$, $C_3^2:Q_8$, $Q_8\times C_9$

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

Order 36: $C_{36}$ x 6, $C_3:C_{12}$ x 2, $C_9:C_4$ x 2, $C_3\times C_{12}$

Order 27: $C_3\times C_9$ x 3

Order 24: $C_3:Q_8$, $C_3\times Q_8$

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

Order 12: $C_{12}$ x 5, $C_3:C_4$ x 2

Order 9: $C_9$ x 4, $C_3^2$

Order 8: $Q_8$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_{36}.C_{18}$

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

Order 216: $C_{12}.C_{18}$, $C_{36}.C_6$

Order 162: $C_9:C_{18}$

Order 108: $C_3\times C_{36}$ x 3, $C_3:C_{36}$, $C_9:C_{12}$

Order 81: $C_9:C_9$

Order 72: $C_9:Q_8$, $C_3^2:Q_8$, $Q_8\times C_9$

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

Order 36: $C_{36}$ x 5, $C_3\times C_{12}$, $C_3:C_{12}$, $C_9:C_4$

Order 27: $C_3\times C_9$ x 3

Order 24: $C_3:Q_8$, $C_3\times Q_8$

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

Order 12: $C_{12}$ x 4, $C_3:C_4$

Order 9: $C_9$ x 4, $C_3^2$

Order 8: $Q_8$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_{36}.C_{18}$ ($C_1$)

Order 324: $C_9:C_{36}$ ($C_2$) x 2, $C_9:C_{36}$ ($C_2$)

Order 216: $C_{36}.C_6$ ($C_3$)

Order 162: $C_9:C_{18}$ ($C_2^2$)

Order 108: $C_9:C_{12}$ ($C_6$) x 2, $C_3\times C_{36}$ ($C_6$), $C_3\times C_{36}$ ($S_3$)

Order 81: $C_9:C_9$ ($Q_8$)

Order 72: $C_9:Q_8$ ($C_9$)

Order 54: $C_3\times C_{18}$ ($C_2\times C_6$), $C_3\times C_{18}$ ($D_6$)

Order 36: $C_9:C_4$ ($C_{18}$) x 2, $C_3\times C_{12}$ ($C_3\times S_3$), $C_{36}$ ($C_{18}$)

Order 27: $C_3\times C_9$ ($C_3:Q_8$), $C_3\times C_9$ ($C_3\times Q_8$)

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

Order 12: $C_{12}$ ($C_9:C_6$), $C_{12}$ ($S_3\times C_9$)

Order 9: $C_3^2$ ($C_3^2:Q_8$), $C_9$ ($Q_8\times C_9$)

Order 6: $C_6$ ($C_{18}:C_6$), $C_6$ ($S_3\times C_{18}$)

Order 4: $C_4$ ($C_9:C_{18}$)

Order 3: $C_3$ ($C_{36}.C_6$), $C_3$ ($C_{12}.C_{18}$)

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

Order 1: $C_1$ ($C_{36}.C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_{36}.C_{18}$ ($C_1$)

Order 324: $C_9:C_{36}$ ($C_2$), $C_9:C_{36}$ ($C_2$)

Order 216: $C_{36}.C_6$ ($C_3$)

Order 162: $C_9:C_{18}$ ($C_2^2$)

Order 108: $C_9:C_{12}$ ($C_6$), $C_3\times C_{36}$ ($C_6$), $C_3\times C_{36}$ ($S_3$)

Order 81: $C_9:C_9$ ($Q_8$)

Order 72: $C_9:Q_8$ ($C_9$)

Order 54: $C_3\times C_{18}$ ($C_2\times C_6$), $C_3\times C_{18}$ ($D_6$)

Order 36: $C_3\times C_{12}$ ($C_3\times S_3$), $C_{36}$ ($C_{18}$), $C_9:C_4$ ($C_{18}$)

Order 27: $C_3\times C_9$ ($C_3:Q_8$), $C_3\times C_9$ ($C_3\times Q_8$)

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

Order 12: $C_{12}$ ($C_9:C_6$), $C_{12}$ ($S_3\times C_9$)

Order 9: $C_3^2$ ($C_3^2:Q_8$), $C_9$ ($Q_8\times C_9$)

Order 6: $C_6$ ($C_{18}:C_6$), $C_6$ ($S_3\times C_{18}$)

Order 4: $C_4$ ($C_9:C_{18}$)

Order 3: $C_3$ ($C_{36}.C_6$), $C_3$ ($C_{12}.C_{18}$)

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

Order 1: $C_1$ ($C_{36}.C_{18}$)

Series

Derived series

$C_{36}.C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_1$

Chief series

$C_{36}.C_{18}$

$\rhd$

$C_9:C_{36}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{36}.C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

Supergroups

This group is a maximal subgroup of 23 larger groups in the database.

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $93 \times 93$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.