Abstract group 144.40: $D_{36}:C_2$

• Label: 144.40
• Order: \( 2^{4} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$D_{36}:C_2$
Order:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2\times D_{36}:C_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	39	2	40	6	6	8	18	24	144
Conjugacy classes	1	4	1	5	3	3	4	9	12	42
Divisions	1	4	1	4	2	1	2	2	2	19
Autjugacy classes	1	3	1	3	2	1	2	2	2	17

Dimension	1	2	4	6	8	24
Irr. complex chars.	8	34	0	0	0	0	42
Irr. rational chars.	8	4	1	4	1	1	19

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$1008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	24
Arbitrary	2	4	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{18}=[a,b]=[b,c]=1, c^{a}=b^{2}c^{17} \rangle$
Permutation group:	Degree $17$ $\langle(2,4)(3,7)(5,6)(8,9)(10,11)(12,13)(14,16)(15,17), (10,12,14,15)(11,13,16,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 36 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{37})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{36}$ $\,\rtimes\,$ $C_2$	$(C_9:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_9:D_4)$ $\,\rtimes\,$ $C_2$ (2)	$(C_4\times D_9)$ $\,\rtimes\,$ $C_2$ (2)	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $D_6$ (2)	$C_4$ . $D_{18}$ (2)	$D_{18}$ . $C_2^2$ (2)	$C_2^2$ . $D_{18}$	all 13

Elements of the group are displayed as matrices in $\GL_{2}(\F_{37})$.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 219 subgroups in 60 conjugacy classes, 29 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $D_{18}$
Commutator:	$G' \simeq$ $C_{18}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{36}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{36}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$

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 144: $D_{36}:C_2$

Order 72: $C_9:D_4$ x 2, $C_4\times D_9$ x 2, $C_2\times C_{36}$, $D_{36}$, $C_9:Q_8$

Order 48: $D_{12}:C_2$

Order 36: $D_{18}$ x 2, $C_{36}$ x 2, $C_9:C_4$ x 2, $C_2\times C_{18}$

Order 24: $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$

Order 18: $C_{18}$ x 2, $D_9$ x 2

Order 16: $D_4:C_2$

Order 12: $D_6$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 9: $C_9$

Order 8: $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 2, $S_3$ x 2

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 144: $D_{36}:C_2$

Order 72: $C_2\times C_{36}$, $C_9:D_4$, $D_{36}$, $C_4\times D_9$, $C_9:Q_8$

Order 48: $D_{12}:C_2$

Order 36: $C_{36}$ x 2, $C_2\times C_{18}$, $D_{18}$, $C_9:C_4$

Order 24: $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$

Order 18: $C_{18}$ x 2, $D_9$

Order 16: $D_4:C_2$

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 9: $C_9$

Order 8: $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 2, $S_3$

Order 4: $C_4$ x 3, $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 144: $D_{36}:C_2$ ($C_1$)

Order 72: $C_9:D_4$ ($C_2$) x 2, $C_4\times D_9$ ($C_2$) x 2, $C_2\times C_{36}$ ($C_2$), $D_{36}$ ($C_2$), $C_9:Q_8$ ($C_2$)

Order 36: $D_{18}$ ($C_2^2$) x 2, $C_{36}$ ($C_2^2$) x 2, $C_9:C_4$ ($C_2^2$) x 2, $C_2\times C_{18}$ ($C_2^2$)

Order 24: $C_2\times C_{12}$ ($S_3$)

Order 18: $C_{18}$ ($C_2^3$)

Order 12: $C_{12}$ ($D_6$) x 2, $C_2\times C_6$ ($D_6$)

Order 9: $C_9$ ($D_4:C_2$)

Order 8: $C_2\times C_4$ ($D_9$)

Order 6: $C_6$ ($C_2\times D_6$)

Order 4: $C_4$ ($D_{18}$) x 2, $C_2^2$ ($D_{18}$)

Order 3: $C_3$ ($D_{12}:C_2$)

Order 2: $C_2$ ($C_2\times D_{18}$)

Order 1: $C_1$ ($D_{36}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 144: $D_{36}:C_2$ ($C_1$)

Order 72: $C_2\times C_{36}$ ($C_2$), $C_9:D_4$ ($C_2$), $D_{36}$ ($C_2$), $C_4\times D_9$ ($C_2$), $C_9:Q_8$ ($C_2$)

Order 36: $C_{36}$ ($C_2^2$) x 2, $C_2\times C_{18}$ ($C_2^2$), $D_{18}$ ($C_2^2$), $C_9:C_4$ ($C_2^2$)

Order 24: $C_2\times C_{12}$ ($S_3$)

Order 18: $C_{18}$ ($C_2^3$)

Order 12: $C_{12}$ ($D_6$) x 2, $C_2\times C_6$ ($D_6$)

Order 9: $C_9$ ($D_4:C_2$)

Order 8: $C_2\times C_4$ ($D_9$)

Order 6: $C_6$ ($C_2\times D_6$)

Order 4: $C_4$ ($D_{18}$) x 2, $C_2^2$ ($D_{18}$)

Order 3: $C_3$ ($D_{12}:C_2$)

Order 2: $C_2$ ($C_2\times D_{18}$)

Order 1: $C_1$ ($D_{36}:C_2$)

Series

Derived series

$D_{36}:C_2$

$\rhd$

$C_{18}$

$\rhd$

$C_1$

Chief series

$D_{36}:C_2$

$\rhd$

$C_2\times C_{36}$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{36}:C_2$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 53 larger groups in the database.

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.