Abstract group 419904.ln: $D_9\wr C_2^2$

• Label: 419904.ln
• Order: \( 2^{6} \cdot 3^{8} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$D_9\wr C_2^2$
Order:	\(419904\)\(\medspace = 2^{6} \cdot 3^{8} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$D_9\wr A_4.C_6$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \)
Composition factors:	$C_2$ x 6, $C_3$ x 8
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	9	12	18	36
Elements	1	10971	80	96228	18432	6480	34992	147744	104976	419904
Conjugacy classes	1	9	6	6	20	168	3	192	9	414
Divisions	1	9	6	6	20	56	3	64	3	168
Autjugacy classes	1	5	4	2	8	23	1	20	1	65

Dimension	1	2	4	8	16	24	32	48	64	96	192
Irr. complex chars.	8	6	2	80	100	0	164	0	54	0	0	414
Irr. rational chars.	8	6	2	20	7	20	2	31	0	54	18	168

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{2}=c^{4}=d^{18}=e^{18}=f^{9}=g^{9}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,34,3,36,2,35)(4,31,17,7)(5,33,16,8)(6,32,18,9)(10,26,12,25,11,27)(13,24,15,23,14,22) \!\cdots\! \rangle$
Transitive group:	36T29121	36T29154			more information
Direct product:	not computed
Semidirect product:	$C_9^4$ $\,\rtimes\,$ $(C_2\wr C_2^2)$				more information
Trans. wreath product:	$D_9$ $\ \wr\ $$C_2^2$
Possibly split product:	$(C_9:D_9^3)$ . $D_4$ (3)	$(D_9^2\wr C_2)$ . $C_2$ (3)	$(C_9:D_9^3)$ . $C_2^3$	$C_3^4$ . $(S_3\wr C_2^2)$	all 12

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 29 normal subgroups (8 characteristic).

Characteristic subgroups are shown in this color. Normal (but not 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:	$P_{ 3 } \simeq$ $C_9^4$

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 1728) are shown, as well as all normal subgroups of any index.

Order 419904: $D_9\wr C_2^2$

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

Order 104976: $C_9^4:(C_2\times D_4)$ x 6, $D_9^2.D_9^2$ x 3, $C_9^4:D_4:C_2$ x 3, $C_9^2\wr C_2.D_4$ x 3, $C_9^2\wr C_2.D_4$ x 3, $C_9^4.C_2^4$

Order 52488: $C_9^4:D_4$ x 9, $C_9^4:(C_2\times C_4)$ x 6, $C_9^4.C_2^3$ x 5, $C_9:D_9^3$ x 5, $C_9^4.(C_2\times C_4)$ x 3, $C_9^4:D_4$ x 3, $C_9\wr D_4$ x 3, $C_9^4.Q_8$

Order 34992: $(C_3\times D_9^2):D_{18}$ x 3, $(C_3\times C_9^3).C_2^4$

Order 26244: $C_9^4.C_2^2$ x 11, $C_9\wr C_2^2$ x 5, $C_9^4.C_4$ x 3, $C_9\wr C_4$ x 3, $C_9\times D_9\times C_9:D_9$ x 2, $D_9\times C_9^3:C_2$, $C_9^2\times D_9^2$, $C_9\times C_9^3.C_2^2$

Order 23328: $(S_3\times D_9)\wr C_2$ x 3

Order 17496: $(C_3\times C_9^3).C_2^3$ x 7, $C_9^3.D_6.C_2$ x 5, $C_9^3.C_6.C_2^2$ x 5, $C_3:D_9^3$ x 5, $C_9\times D_9^2:S_3$ x 3, $C_9^3.D_{12}$ x 3, $C_9^3:(C_4\times S_3)$ x 3, $(C_3\times C_9^2):D_{36}$ x 3, $C_3\times C_9^3:D_4$ x 3

Order 13122: $C_9^4.C_2$ x 3, $C_9^2\wr C_2$ x 3, $C_9\times C_9^3:C_2$ x 2, $C_9^2\times C_9\times D_9$

Order 11664: $C_9^2:(S_3^2:C_2^2)$ x 6, $D_9^2:D_{18}$ x 3, $C_3^2.C_{18}^2:C_4$ x 3, $S_3^2:D_9^2$ x 3, $C_9^2:(C_6^2:C_4)$ x 3, $C_9^2:(S_3^2:C_4)$ x 3, $D_9^2:S_3^2$ x 3, $C_9^3.C_2^4$, $C_9^2.D_6^2$, $C_3^4.D_6^2$, $(C_3^2\times C_9).D_6^2$

Order 8748: $C_9^3.D_6$ x 20, $C_9^3:D_6$ x 12, $(C_3\times C_9^3).C_2^2$ x 9, $C_9\times D_9\times C_3:D_9$ x 6, $C_3\times C_9:D_9^2$ x 5, $C_9^3.C_6.C_2$ x 3, $(C_3\times C_9^2):C_{36}$ x 3, $C_9^3:C_{12}$ x 3, $D_9\times C_9\times C_3:D_9$ x 2, $C_9\times S_3\times C_9:D_9$ x 2, $C_9\times (C_3\times C_9^2):C_2^2$ x 2, $C_3\times D_9\times C_9:D_9$ x 2, $C_3\times C_9\times D_9^2$ x 2, $S_3\times C_9^3:C_2$, $D_9\times C_3\times C_9:D_9$, $C_9^2\times S_3\times D_9$, $C_9^2.(C_{18}\times S_3)$, $(C_3\times C_9).C_{18}^2$

Order 6561: $C_9^4$

Order 5832: $D_9^3$ x 21, $C_9^2:\SOPlus(4,2)$ x 18, $C_3^3.S_3^3$ x 12, $(C_3\times C_9).S_3^3$ x 7, $C_9^3:D_4$ x 6, $(C_3^2\times C_9^2):(C_2\times C_4)$ x 6, $(C_3^2\times C_9^2).C_2^3$ x 5, $C_3^3.S_3^3$ x 4, $C_2\times C_9^3.C_2^2$ x 3, $(C_3\times C_9^2).D_6.C_2$ x 3, $D_9^2:C_{18}$ x 3, $C_9^2:C_4\times D_9$ x 3, $C_9^2:D_{36}$ x 3, $(C_9\times S_3)\wr C_2$ x 3, $C_9:D_9.S_3^2$ x 3, $C_3^2\wr C_2.S_3^2$ x 3, $C_9^2:\SOPlus(4,2)$ x 3, $(C_3\times C_9^2):D_{12}$ x 3, $(C_3\times D_9)\wr C_2$ x 3, $C_9^3.C_2^3$ x 2, $C_9^2.(C_6\times S_3).C_2$ x 2, $C_9^2.(C_2\times S_3^2)$ x 2, $C_9.(C_9\times S_3^2).C_2$ x 2, $C_{18}\times D_9^2$, $C_{18}\times D_9\times D_9$, $C_3^4.C_6^2.C_2$, $(C_3^3\times C_9).D_6.C_2$, $C_9^2:\PSU(3,2)$

Order 5184: $S_3\wr C_2^2$

Order 4374: $C_9\times (C_3\times C_9^2):C_2$ x 7, $C_9^3:S_3$ x 7, $C_9^3:C_6$ x 7, $C_9^3.S_3$ x 6, $C_9^3.C_6$ x 5, $C_3\times C_9\times C_9\times D_9$ x 3, $C_9^2\times C_3:D_9$ x 2, $C_9\times C_9\times C_3:D_9$ x 2, $C_3\times C_9\times C_9:D_9$ x 2, $C_9^2\times C_9\times S_3$, $C_9^2\times C_3\times D_9$, $C_9\times C_3\times C_9\times D_9$, $C_3\times C_9^3:C_2$, $(C_3\times C_9^3).C_2$

Order 3888: $D_9^2:D_6$ x 3, $S_3^2:(S_3\times D_9)$ x 3, $C_3^3.D_6^2$, $C_3^2.S_3^3.C_2$, $C_2\times C_3^2.S_3^3$, $(C_3\times C_9).D_6^2$

Order 2916: $C_9^3.C_2^2$ x 69, $(C_3\times C_9^2).D_6$ x 40, $C_9\times D_9^2$ x 29, $C_9:D_9^2$ x 13, $C_9^2.C_3:S_3.C_2$ x 12, $C_9^2:S_3^2$ x 12, $C_3^2:D_9^2$ x 12, $C_9^2.S_3^2$ x 11, $(C_3^3\times C_9).D_6$ x 11, $C_9\times D_9\times D_9$ x 10, $C_9\times S_3\times C_3:D_9$ x 6, $C_9\times D_9^2$ x 6, $C_3\times D_9\times C_3:D_9$ x 6, $(C_3^2\times C_9).C_6^2$ x 4, $D_9\times C_9:D_9$ x 3, $D_9\times C_3\times C_3:D_9$ x 3, $C_3\times C_9\times S_3\times D_9$ x 3, $C_9^2:C_{36}$ x 3, $C_9^3:C_4$ x 3, $(C_3^2\times C_9):C_{36}$ x 3, $(C_3\times C_9^2):C_{12}$ x 3, $S_3\times C_9\times C_3:D_9$ x 2, $C_{18}\times C_9:D_9$ x 2, $C_9\times D_9\times C_3:S_3$ x 2, $C_3\times S_3\times C_9:D_9$ x 2, $C_3\times (C_3\times C_9^2):C_2^2$ x 2, $D_{18}\times C_9^2$ x 2, $S_3\times C_3\times C_9:D_9$, $D_9\times C_9\times C_3:S_3$, $C_9^2\times S_3^2$, $C_9^2.C_6^2$, $C_9\times C_{18}\times D_9$, $C_9\times C_9\times S_3^2$, $C_9\times C_3\times S_3\times D_9$, $C_9:D_9\times D_9$, $C_9.(C_9\times S_3^2)$, $C_3^4.C_6^2$, $C_3^2\times D_9^2$, $C_2\times C_9^3:C_2$, $C_9^2:D_{18}$

Order 2592: $D_{18}\wr C_2$ x 3, $S_3^2\wr C_2$ x 3, $C_3:S_3^3:C_4$ x 3, $C_3:S_3^3:C_2^2$

Order 2187: $C_3\times C_9^3$ x 6

Order 1944: $S_3\times D_9^2$ x 51, $C_6:D_9^2$ x 9, $C_9:S_3^3$ x 6, $S_3\times C_9\times D_{18}$ x 6, $D_9^2:S_3$ x 6, $C_9:S_3^3$ x 5, $C_9:S_3^3$ x 3, $C_3^2.S_3^3$ x 3, $C_3^2.S_3^3$ x 3, $(C_3^2\times C_9):D_{12}$ x 3, $C_3^3:D_{36}$ x 3, $C_9\times S_3^2:S_3$ x 3, $C_3\times S_3^2:D_9$ x 3, $C_3^3:(C_4\times D_9)$ x 3, $C_6:D_9^2$ x 3, $S_3\times C_9:D_{18}$ x 3, $C_6\times D_9^2$ x 3, $C_9^2:D_{12}$ x 3, $D_9^2:C_6$ x 3, $C_9^2:C_4\times S_3$ x 3, $C_3^2.S_3^3$, $C_9\times S_3^3$

Order 81: $C_3^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 419904: $D_9\wr C_2^2$ ($C_1$)

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

Order 104976: $C_9^2\wr C_2.D_4$ ($C_2^2$) x 3, $C_9^4:(C_2\times D_4)$ ($C_2^2$) x 3, $C_9^4.C_2^4$ ($C_2^2$)

Order 52488: $C_9^4:(C_2\times C_4)$ ($D_4$) x 3, $C_9:D_9^3$ ($D_4$) x 3, $C_9:D_9^3$ ($C_2^3$)

Order 26244: $C_9^4.C_2^2$ ($C_2\times D_4$) x 3

Order 13122: $C_9^4.C_2$ ($C_2^2\wr C_2$)

Order 6561: $C_9^4$ ($C_2\wr C_2^2$)

Order 81: $C_3^4$ ($S_3\wr C_2^2$)

Order 1: $C_1$ ($D_9\wr C_2^2$)

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 7 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 $414 \times 414$ 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 $168 \times 168$ rational character table (warning: may be slow to load).