Abstract group 1259712.jj: $D_9\wr C_4.C_3$

• Label: 1259712.jj
• Order: \( 2^{6} \cdot 3^{9} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$D_9\wr C_4.C_3$
Order:	\(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$D_9\wr C_4.C_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \)
Composition factors:	$C_2$ x 6, $C_3$ x 9
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36
Elements	1	10323	242	43740	60390	104976	19440	192456	303264	209952	314928	1259712
Conjugacy classes	1	6	7	4	27	2	63	11	85	4	9	219
Divisions	1	6	6	3	21	1	58	5	70	1	4	176
Autjugacy classes	1	6	7	3	24	1	48	8	60	2	6	166

Dimension	1	2	4	8	16	24	32	48	64	96	192
Irr. complex chars.	24	6	9	36	27	12	6	33	0	46	20	219
Irr. rational chars.	4	8	7	15	19	12	10	31	3	47	20	176

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{2}=c^{18}=d^{18}=e^{18}=f^{9}=[c,f]=1, a^{12}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,22,7,5,15,10,31,28,3,23,9,4,13,12,32,29,2,24,8,6,14,11,33,30)(16,27,35,21,17,26,36,20) \!\cdots\! \rangle$
Transitive group:	36T39258	36T39270			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(D_9\wr C_4)$ . $C_3$	$C_3^5$ . $(S_3\wr C_4)$	$(D_9^2\wr C_2)$ . $C_6$	$C_3^4$ . $(S_3^4:C_{12})$	all 26

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 14936302 subgroups in 14238 conjugacy classes, 29 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_9:D_9^3$
Frattini:	a subgroup isomorphic to $C_3^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^4.C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 243 (order at least 5184) are shown, as well as all normal subgroups of any index.

Order 1259712: $D_9\wr C_4.C_3$

Order 629856: $D_9^2\wr C_2.C_3$, $C_9:D_9^3.C_{12}$, $C_9:D_9^3.C_{12}$

Order 419904: $D_9\wr C_4$

Order 314928: $C_9:D_9^3.C_6$ x 2, $C_9^4.C_6.C_2^3$, $C_9^4.(C_6\times D_4)$, $C_9^4.(C_3\times \OD_{16})$, $C_9^4.(C_6\times D_4)$, $C_9^2.C_{18}^2:C_{12}$

Order 209952: $D_9^2\wr C_2$, $C_9^4.C_2^2.D_4$, $C_9:D_9^3.C_4$

Order 157464: $C_9^4.C_6.C_2^2$ x 3, $C_9^4.(C_2\times C_{12})$ x 2, $C_9^4.(C_2^2\times C_6)$ x 2, $C_9^3.C_9:C_3.C_2^3$, $C_9^4.C_{24}$, $C_9^4.(C_3\times D_4)$, $C_9^4.(C_2\times C_{12})$, $C_9^4:(C_3\times D_4)$, $C_9^3.C_6.C_6^2$

Order 104976: $C_9^2\wr C_2.D_4$ x 2, $C_9^4.C_2^4$, $C_9^3.C_3^2.C_2^4$, $D_9^2.D_9^2$, $C_9^4:\OD_{16}$, $C_9^4:(C_2\times D_4)$, $C_9^2\wr C_2.D_4$, $C_3:D_9^3:C_6$

Order 78732: $C_9^4.C_6.C_2$ x 6, $(C_3\times C_9^3).C_6^2$ x 3, $C_9^2.C_3^3.C_6^2$ x 2, $C_9^4.C_{12}$ x 2, $C_9^4:(C_2\times C_6)$ x 2, $C_9^4.C_{12}$

Order 69984: $C_3^4.(C_2\times C_6\times S_3^2).C_2$

Order 52488: $C_3^3.C_3^5.C_2^3$ x 5, $C_9^4.C_2^3$ x 4, $C_9^3.C_6^2.C_2$ x 3, $C_9^3.C_3^2.C_2^3$ x 3, $C_9^4.(C_2\times C_4)$ x 2, $C_9:D_9^3$ x 2, $C_9^3:(S_3\times C_{12})$ x 2, $C_3^5.S_3^3$ x 2, $C_9^3.C_9.C_2^3$, $C_9^2.C_3^3.C_6.C_2^2$, $C_3^4.C_3^4.C_2^3$, $C_3^3.C_3^4.C_6.C_2^2$, $C_3^2.C_3^5.C_6.C_2^2$, $C_3\times C_9^3.C_6.C_2^2$, $C_9^4:C_8$, $C_9^4:D_4$, $C_9^4:(C_2\times C_4)$, $C_9^4:D_4$, $C_9^3:(D_4\times C_9)$, $C_9\wr D_4$, $C_9^3:C_6\wr C_2$, $C_9^3:(C_3\times D_{12})$, $C_9^3.C_6^2.C_2$, $C_9^3.C_2.C_6^2$

Order 39366: $C_9^2.C_3^3.C_3.C_6$ x 3, $C_9^4.C_6$, $C_9^3.C_3^2.C_6$, $C_9.C_3^4.C_3^3.C_2$

Order 34992: $C_3^2.C_3^5.C_2^4$ x 2, $C_9^2:(S_3^2:C_{12})$ x 2, $C_9^3.C_6.C_2^3$, $C_2\times C_9^3.C_6.C_2^2$, $(C_3\times D_9^2):D_{18}$, $D_9^3:C_6$, $C_3^2:(C_{18}^2:C_{12})$, $C_9^2.(C_6\times S_3\wr C_2)$, $C_9^2:(C_6^2:C_{12})$, $D_9^2:(C_3\times S_3^2)$, $C_3^5.S_3^2:C_2^2$

Order 26244: $C_9^3.(C_6\times S_3)$ x 13, $C_9^4.C_2^2$ x 9, $C_9^2.C_3^3.C_6.C_2$ x 9, $C_3^3.C_3^5.C_2^2$ x 9, $C_9^2.C_9:C_9.C_2^2$ x 5, $C_9^3:(C_6\times S_3)$ x 5, $C_3\times (C_3\times C_9^2).C_6^2$ x 4, $C_9^3.C_{18}.C_2$ x 3, $C_9.C_3^4.C_6^2$ x 3, $C_9^3.C_6^2$ x 2, $C_9\times C_9^3.C_2^2$ x 2, $C_3^3.C_3^4.C_6.C_2$ x 2, $C_9^3:C_{36}$ x 2, $C_9\wr C_4$ x 2, $C_9^3:(C_2\times C_{18})$ x 2, $C_9\wr C_2^2$ x 2, $(C_3\times C_9^3):C_{12}$ x 2, $C_3\times C_9^3:C_{12}$ x 2, $C_9^3:C_6^2$ x 2, $C_9^4.C_4$, $C_9^2.C_3^2.C_6^2$, $C_3^4.C_3^4.C_2^2$, $C_3^2.C_3^5.C_6.C_2$, $C_3\times C_9^3.C_6.C_2$

Order 23328: $(S_3\times D_9)\wr C_2$

Order 19683: $C_9^4.C_3$

Order 17496: $D_9^3:C_3$ x 17, $(C_3\times C_9^3).C_2^3$ x 12, $C_3^2.C_3^5.C_2^3$ x 11, $C_3^3.C_3^4.C_2^3$ x 5, $C_3^4.S_3^3$ x 4, $C_9^3.C_6.C_2^2$ x 3, $C_9^2.C_3^3.C_2^3$ x 3, $C_3^5.\SOPlus(4,2)$ x 3, $C_9^2.C_9:C_3.C_2^3$ x 2, $C_3.C_3^5.C_6.C_2^2$ x 2, $C_2\times C_9^3.C_6.C_2$ x 2, $C_9^3:(C_4\times S_3)$ x 2, $C_9^3:(C_2\times C_{12})$ x 2, $C_3:D_9^3$ x 2, $(C_9\times D_9^2):C_6$ x 2, $C_9^2:C_6.S_3^2$ x 2, $C_3^4:C_6.S_3^2$ x 2, $C_9^3.D_6.C_2$, $C_9^2.(C_{18}\times S_3).C_2$, $C_9^2.(C_6\times C_{18}).C_2$, $C_9.C_3^4.C_6.C_2^2$, $C_3^4.S_3^3$, $C_3\times C_9^3.C_2^3$, $C_3\times C_9^2.C_3^2.C_2^3$, $C_3\times C_3^3.C_3^3.C_2^3$, $C_3\times C_3^2.C_3^4.C_2^3$, $C_2\times (C_3\times C_9^2).C_6^2$, $(C_3\times C_9^2).C_9.C_2^3$, $(C_3\times D_9^2):C_{18}$, $C_9\times D_9^2:S_3$, $(C_9\times D_9^2):C_6$, $C_9^3.D_{12}$, $(C_3\times C_9^2):D_{36}$, $C_9^3:(C_3\times D_4)$, $C_3\times C_9^3:D_4$, $C_3^5.F_9$, $(C_3^3\times C_9).C_{12}.C_6$, $(C_3^2\times C_9^2):(C_2\times C_{12})$, $C_9^2:S_3^2:C_6$, $C_3^5.\SOPlus(4,2)$, $D_9^2:(S_3\times C_3^2)$

Order 15552: $S_3^4:C_{12}$

Order 13122: $C_9^2.C_9:C_9.C_2$ x 7, $C_9^2.C_3^3.C_6$ x 7, $C_9.C_3^4.C_3.C_6$ x 7, $C_9.(C_3\times C_9:C_9).C_6$ x 7, $C_3\times (C_3\times C_9).C_3^3.C_6$ x 7, $C_9^3.C_3^2.C_2$ x 5, $C_3\times C_9^2.C_3^2.C_6$ x 4, $C_9^2.C_9:C_3.C_6$ x 3, $C_9\times C_9^3:C_2$ x 3, $C_9^2\wr C_2$ x 2, $C_9^4.C_2$, $C_9^3.C_{18}$, $C_3^5.C_3^3.C_2$, $C_3\times C_9^3.C_6$

Order 11664: $C_9^2:(S_3^2:C_4)$ x 2, $C_9^2.D_6^2$, $C_9^2.C_3^2.C_2^4$, $C_3.C_3^5.C_2^4$, $C_2\times C_9^3.C_2^3$, $C_2\times C_3^3.C_3^3.C_2^3$, $C_2\times C_3^2.C_3^4.C_2^3$, $(C_3^2\times C_9).D_6^2$, $D_9^2:D_{18}$, $C_3^2.C_{18}^2:C_4$, $S_3^2:D_9^2$, $C_9^2:(C_6^2:C_4)$, $D_9^2:S_3^2$, $D_9^2:C_6\times S_3$, $C_9^2:(S_3^2:C_2^2)$, $C_3^5.(S_3\times D_4)$

Order 8748: $C_9^3.C_6.C_2$ x 60, $C_9^3:(C_2\times C_6)$ x 37, $C_3^2.C_3^5.C_2^2$ x 22, $C_9^2.C_3^3.C_2^2$ x 17, $C_9^3.D_6$ x 16, $C_9^2.(C_{18}\times S_3)$ x 13, $C_9^3:(C_2\times C_6)$ x 13, $C_9.C_3^4.C_6.C_2$ x 10, $C_3^3.C_3^4.C_2^2$ x 10, $(C_3^3\times C_9).C_6^2$ x 9, $C_9\times (C_3\times C_9^2):C_2^2$ x 7, $C_3\times (C_3^2\times C_9).C_6^2$ x 7, $(C_3\times C_9^3).C_2^2$ x 7, $C_3\times C_9^3.C_2^2$ x 5, $C_9^2:(S_3\times C_{18})$ x 5, $C_9^3:D_6$ x 5, $C_9^2:(C_3\times S_3^2)$ x 5, $C_3^5.S_3^2$ x 5, $C_9\times D_9\times C_3:D_9$ x 4, $C_3\times C_9^2.C_{18}.C_2$ x 4, $(C_3^2\times C_9:C_3).C_6^2$ x 4, $C_9^2.C_3:S_3.C_6$ x 3, $C_3\times C_9^2.(C_6\times S_3)$ x 3, $D_9\times C_3\times C_9:D_9$ x 2, $C_3^5.C_6^2$ x 2, $C_3^4.C_3^3.C_2^2$ x 2, $C_3.C_3^5.C_6.C_2$ x 2, $C_2\times C_9^2.C_3^2.C_6$ x 2, $(C_3\times C_9^2):C_{36}$ x 2, $(C_3\times C_9^2):C_{36}$ x 2, $C_9^3:C_{12}$ x 2, $C_9^3:C_{12}$ x 2, $C_9^3:C_{12}$ x 2, $C_3\times C_9:D_9^2$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $S_3\times (C_3^3.C_3^3):C_2$, $C_9^2.C_9.C_6.C_2$, $C_9\times S_3\times C_9:D_9$, $C_3^2\times C_9:(C_9:C_3):C_2^2$, $C_3\times C_9^2.C_6^2$, $C_3\times C_3^2.C_3^4.C_2^2$, $C_2\times C_9^3.C_6$, $C_2\times (C_3\times C_9).C_3^3.C_6$, $(C_3\times C_9^2).C_{18}.C_2$, $(C_3\times C_9^2).C_6^2$, $(C_3\times C_9).C_{18}^2$

Order 7776: $(D_{18}\times D_{18}):C_6$, $C_3:S_3^3:C_{12}$, $S_3^4:C_6$, $C_3:S_3^3.C_{12}$

Order 6561: $C_3\times C_9^3.C_3$ x 5, $C_3^4.C_3^4$ x 5, $C_9^4$

Order 5832: $C_3^3.S_3^3$ x 43, $D_9^3$ x 17, $(C_3^2\times C_9^2).C_2^3$ x 10, $C_3.C_3^5.C_2^3$ x 8, $C_2\times C_9^2.(C_6\times S_3)$ x 7, $C_9^2.C_6^2.C_2$ x 4, $(C_3^3\times C_9).D_6.C_2$ x 4, $(C_3^2\times C_9).C_6^2.C_2$ x 4, $(C_3\times C_9).S_3^3$ x 4, $C_3^3.S_3^3$ x 4, $C_9^3.C_2^3$ x 3, $C_9^2.C_3^2.C_2^3$ x 3, $C_9.C_3^4.C_2^3$ x 3, $C_3^4.C_6^2.C_2$ x 3, $C_3\times C_3^2.S_3^3$ x 3, $C_9^2:\SOPlus(4,2)$ x 3, $C_9^2.S_3^2.C_2$ x 2, $C_9^2.C_9.C_2^3$ x 2, $C_9\times C_9^2:C_2^3$ x 2, $C_9.(C_3\times C_{18}\times S_3).C_2$ x 2, $C_3^2.C_3^4.C_2^3$ x 2, $C_3\times C_3^2.(C_6\times S_3^2)$ x 2, $C_2\times (C_3^2\times C_9).C_6^2$ x 2, $C_9^2:C_4\times D_9$ x 2, $C_9^3:D_4$ x 2, $S_3\times C_9^2:C_{12}$ x 2, $C_9:D_9.S_3^2$ x 2, $C_3^2\wr C_2.S_3^2$ x 2, $(C_3\times D_9^2):C_6$ x 2, $C_3^4.(S_3\times C_{12})$ x 2, $C_3^3.S_3^3$ x 2, $C_{18}\times D_9^2$, $C_6\times C_9:(C_9:C_3):C_2^2$, $C_3^3.S_3^2:S_3$, $C_3\times C_9.(C_6\times S_3^2)$, $C_2\times C_9^3.C_2^2$, $C_2\times C_9^2.C_{18}.C_2$, $C_2\times C_3^2.C_3^4.C_2^2$, $(C_3^3\times C_9).C_6.C_2^2$, $(C_3\times C_9^2).D_6.C_2$, $D_9^2:C_{18}$, $D_9^2:C_{18}$, $C_9^2:D_{36}$, $C_9^2:F_9$, $(C_9\times S_3^2):C_{18}$, $(C_9\times S_3)\wr C_2$, $(C_3^2\times C_9^2):(C_2\times C_4)$, $C_9^2:\SOPlus(4,2)$, $C_9^2:(C_3\times D_{12})$, $(C_3\times D_9)\wr C_2$, $C_3\times D_9^2:C_6$, $(C_3^3\times S_3^2).S_3$, $C_9:C_3\wr D_4$, $C_3^2.C_3\wr D_4$, $C_3^5.D_{12}$

Order 5184: $S_3\wr C_4$

Order 243: $C_3^5$

Order 81: $C_3^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1259712: $D_9\wr C_4.C_3$ ($C_1$)

Order 629856: $D_9^2\wr C_2.C_3$ ($C_2$), $C_9:D_9^3.C_{12}$ ($C_2$), $C_9:D_9^3.C_{12}$ ($C_2$)

Order 419904: $D_9\wr C_4$ ($C_3$)

Order 314928: $C_9^4.C_6.C_2^3$ ($C_4$), $C_9:D_9^3.C_6$ ($C_4$), $C_9^4.(C_6\times D_4)$ ($C_2^2$)

Order 209952: $D_9^2\wr C_2$ ($C_6$), $C_9^4.C_2^2.D_4$ ($C_6$), $C_9:D_9^3.C_4$ ($C_6$)

Order 157464: $C_9^4.(C_2\times C_{12})$ ($D_4$), $C_9^4.(C_2^2\times C_6)$ ($D_4$), $C_9^4.(C_2^2\times C_6)$ ($C_2\times C_4$)

Order 104976: $C_9^4.C_2^4$ ($C_{12}$), $C_9^2\wr C_2.D_4$ ($C_{12}$), $C_9^4:(C_2\times D_4)$ ($C_2\times C_6$)

Order 78732: $C_9^4.C_6.C_2$ ($C_2^2:C_4$)

Order 52488: $C_9^4:(C_2\times C_4)$ ($C_3\times D_4$), $C_9:D_9^3$ ($C_2\times C_{12}$), $C_9:D_9^3$ ($C_3\times D_4$)

Order 39366: $C_9^4.C_6$ ($C_2^3:C_4$)

Order 26244: $C_9^4.C_2^2$ ($C_2^2:C_{12}$)

Order 19683: $C_9^4.C_3$ ($C_2\wr C_4$)

Order 13122: $C_9^4.C_2$ ($C_2^3:C_{12}$)

Order 6561: $C_9^4$ ($C_2^4:C_{12}$)

Order 243: $C_3^5$ ($S_3\wr C_4$)

Order 81: $C_3^4$ ($S_3^4:C_{12}$)

Order 1: $C_1$ ($D_9\wr C_4.C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $219 \times 219$ 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 $176 \times 176$ rational character table (warning: may be slow to load).