Abstract group 144.41: $D_4\times D_9$

• Label: 144.41
• Order: \( 2^{4} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$D_4\times D_9$
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	59	2	20	10	6	4	30	12	144
Conjugacy classes	1	7	1	2	3	3	1	9	3	30
Divisions	1	7	1	2	3	1	1	3	1	20
Autjugacy classes	1	4	1	2	2	1	1	2	1	15

Dimension	1	2	4	6	12
Irr. complex chars.	8	18	4	0	0	30
Irr. rational chars.	8	6	1	4	1	20

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$1008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	12
Arbitrary	4	4	8

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{36}=[a,b]=1, c^{a}=c^{17}, c^{b}=c^{19} \rangle$
Permutation group:	Degree $13$ $\langle(2,4)(3,7)(5,6)(8,9), (11,13), (10,11)(12,13), (10,12)(11,13), (1,2,5,7,9,8,3,6,4), (1,3,7)(2,6,9)(4,8,5)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 16 \\ 12 & 23 \end{array}\right), \left(\begin{array}{rr} 28 & 9 \\ 9 & 28 \end{array}\right), \left(\begin{array}{rr} 26 & 9 \\ 9 & 8 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 24 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Transitive group:	36T135				more information
Direct product:	$D_4$ $\, \times\, $ $D_9$
Semidirect product:	$D_{36}$ $\,\rtimes\,$ $C_2$	$C_4$ $\,\rtimes\,$ $D_{18}$	$D_{18}$ $\,\rtimes\,$ $C_2^2$ (2)	$C_2^2$ $\,\rtimes\,$ $D_{18}$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $D_6$	$D_{18}$ . $C_2^2$	$C_{18}$ . $C_2^3$	$(C_3\times D_4)$ . $S_3$	all 8

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

Homology

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

Subgroups

There are 355 subgroups in 81 conjugacy classes, 31 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_{18}$
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:	$\operatorname{Fit} \simeq$ $D_4\times C_9$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4\times D_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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

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Classes of subgroups up to conjugation

Order 144: $D_4\times D_9$

Order 72: $C_9:D_4$ x 2, $C_2\times D_{18}$ x 2, $D_{36}$, $C_4\times D_9$, $D_4\times C_9$

Order 48: $S_3\times D_4$

Order 36: $D_{18}$ x 7, $C_2\times C_{18}$ x 2, $C_{36}$, $C_9:C_4$

Order 24: $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 18: $D_9$ x 4, $C_{18}$ x 3

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_9$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 144: $D_4\times D_9$

Order 72: $C_9:D_4$, $D_{36}$, $C_4\times D_9$, $C_2\times D_{18}$, $D_4\times C_9$

Order 48: $S_3\times D_4$

Order 36: $D_{18}$ x 3, $C_2\times C_{18}$, $C_{36}$, $C_9:C_4$

Order 24: $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_2\times D_6$, $C_3\times D_4$

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

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 3, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_9$

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

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 144: $D_4\times D_9$ ($C_1$)

Order 72: $C_9:D_4$ ($C_2$) x 2, $C_2\times D_{18}$ ($C_2$) x 2, $D_{36}$ ($C_2$), $C_4\times D_9$ ($C_2$), $D_4\times C_9$ ($C_2$)

Order 36: $D_{18}$ ($C_2^2$) x 3, $C_2\times C_{18}$ ($C_2^2$) x 2, $C_{36}$ ($C_2^2$), $C_9:C_4$ ($C_2^2$)

Order 24: $C_3\times D_4$ ($S_3$)

Order 18: $D_9$ ($D_4$) x 2, $C_{18}$ ($C_2^3$)

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

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

Order 8: $D_4$ ($D_9$)

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

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

Order 3: $C_3$ ($S_3\times D_4$)

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

Order 1: $C_1$ ($D_4\times D_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 144: $D_4\times D_9$ ($C_1$)

Order 72: $C_9:D_4$ ($C_2$), $D_{36}$ ($C_2$), $C_4\times D_9$ ($C_2$), $C_2\times D_{18}$ ($C_2$), $D_4\times C_9$ ($C_2$)

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

Order 24: $C_3\times D_4$ ($S_3$)

Order 18: $C_{18}$ ($C_2^3$), $D_9$ ($D_4$)

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

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

Order 8: $D_4$ ($D_9$)

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

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

Order 3: $C_3$ ($S_3\times D_4$)

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

Order 1: $C_1$ ($D_4\times D_9$)

Series

Derived series

$D_4\times D_9$

$\rhd$

$C_{18}$

$\rhd$

$C_1$

Chief series

$D_4\times D_9$

$\rhd$

$C_2\times D_{18}$

$\rhd$

$D_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4\times D_9$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 46 larger groups in the database.

This group is a maximal quotient of 63 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.