Abstract group 314928.qa: $C_9:D_9^3.C_6$

• Label: 314928.qa
• Order: \( 2^{4} \cdot 3^{9} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_9:D_9^3.C_6$
Order:	\(314928\)\(\medspace = 2^{4} \cdot 3^{9} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.S_3\wr D_4$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \)
Composition factors:	$C_2$ x 4, $C_3$ 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	9	12	18	36
Elements	1	7047	242	5832	25758	19440	46656	104976	104976	314928
Conjugacy classes	1	5	11	4	20	163	12	42	12	270
Divisions	1	5	10	2	15	154	4	32	4	227
Autjugacy classes	1	3	7	1	10	32	3	15	3	75

Dimension	1	2	4	8	12	16	24	32	48
Irr. complex chars.	24	6	48	12	16	9	28	0	127	270
Irr. rational chars.	4	8	16	18	12	9	30	3	127	227

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:	${\langle a, b, c, d, e, f \mid b^{2}=c^{18}=d^{9}=e^{9}=f^{9}=[d,e]=[d,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,17,32,10,25,28,8,35,13,4,20,23,3,18,33,11,27,29,7,36,14,6,19,22,2,16,31,12,26,30,9,34,15,5,21,24) \!\cdots\! \rangle$
Transitive group:	36T27826				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_9:D_9^3)$ . $C_6$	$C_9^2$ . $(D_9^2:C_{12})$ (2)	$(C_9^4.C_2^2)$ . $C_{12}$ (2)	$C_9^4$ . $(C_2^2:C_{12})$	all 23

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:	not computed
Commutator length:	$1$

Subgroups

There are 7985302 subgroups in 19337 conjugacy classes, 39 normal (13 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:	not computed
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 108 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 314928: $C_9:D_9^3.C_6$

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

Order 104976: $C_9^2\wr C_2.D_4$

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

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

Order 39366: $C_9^2.C_9:C_9.C_6$ x 2, $C_9^4.C_6$, $C_9^2.C_3^3.C_3.C_6$, $(C_3\times C_9^2).C_3^3.C_6$

Order 34992: $C_9^2:(S_3^2:C_{12})$ x 2

Order 26244: $C_9^3:(C_6\times S_3)$ x 6, $C_3^3.C_3^5.C_2^2$ x 5, $C_9^3.(C_6\times S_3)$ x 4, $C_9^4.C_2^2$ x 3, $C_9^3:C_{36}$ 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, $C_3^4.C_3^4.C_2^2$

Order 19683: $C_9^4.C_3$

Order 17496: $C_3^4.S_3^3$ x 4, $C_9^3:(C_4\times S_3)$ x 2, $C_9^3:(C_2\times C_{12})$ x 2, $C_3:D_9^3$ x 2, $D_9^3:C_3$ x 2, $C_9^2:C_6.S_3^2$ x 2, $C_3^4:C_6.S_3^2$ x 2

Order 13122: $C_9^2.C_9:C_9.C_2$ x 10, $C_9^3.C_3^2.C_2$ x 9, $(C_3^3\times C_9).C_3^3.C_2$ x 6, $C_9^2.C_9:C_3.C_6$ x 4, $C_9^2\wr C_2$ x 4, $C_9^2.C_9.C_3.C_6$ x 3, $C_9.C_3^4.C_3.C_6$ x 2, $C_3\times C_9^2.C_3^2.C_6$ x 2, $(C_3\times C_9).C_9:C_9.C_6$ x 2, $C_9^4.C_2$, $C_3\times C_9^2.C_9.C_6$

Order 11664: $C_9^2:(S_3^2:C_4)$ x 2

Order 8748: $C_9^3.C_6.C_2$ x 22, $C_3^2.C_3^5.C_2^2$ x 9, $C_9^3.D_6$ x 8, $C_3^3.C_3^4.C_2^2$ x 6, $C_9^2:(S_3\times C_{18})$ x 6, $C_9^3:D_6$ x 6, $C_9^3:(C_2\times C_6)$ x 6, $C_9^2:(C_3\times S_3^2)$ x 6, $C_3^5.S_3^2$ x 6, $C_9^2.C_3^3.C_2^2$ x 5, $(C_3\times C_9^3).C_2^2$ 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, $C_9^3:(C_2\times C_6)$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $(C_3^2\times C_9^2):C_{12}$ x 2, $C_3^4.C_3^3.C_2^2$

Order 6561: $C_3\times C_9^3.C_3$ x 9, $C_3^4.C_3^4$ x 9, $C_9^4$

Order 5832: $C_3^3.S_3^3$ x 6, $C_3^3.S_3^3$ x 4, $C_9^2:C_4\times D_9$ x 2, $D_9^3$ 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, $C_3^4.(S_3\times C_{12})$ x 2, $C_3^3.S_3^3$ x 2

Order 4374: $C_9^3.C_6$ x 147, $C_9.(S_3\times C_9:C_9)$ x 30, $C_9^2.C_3^3.C_2$ x 24, $(C_3\times C_9).C_3^3.C_6$ x 18, $C_9^2.C_9.C_6$ x 16, $C_9^2.C_3^2.C_6$ x 14, $(C_3\times C_9).C_9:C_3.C_6$ x 14, $C_3\times (C_3\times C_9:(C_9:C_3)):C_2$ x 12, $C_9^3.S_3$ x 9, $C_9^3:S_3$ x 9, $(C_3\times C_9:C_9).C_3.C_6$ x 8, $C_9^3:C_6$ x 8, $C_9.C_9:C_9.C_6$ x 4, $C_3^4.C_3^3.C_2$ x 3, $C_3\times (C_3^3.C_3^3):C_2$ x 2, $C_9\times C_9\times C_3:D_9$, $C_3^5.C_3.C_6$

Order 3888: $D_9^2:C_{12}$ x 2, $\GL(2,3).C_2^6$

Order 2916: $C_9^2.(C_6\times S_3)$ x 60, $C_9^3.C_2^2$ x 22, $C_3^2.C_3^4.C_2^2$ x 22, $(C_3\times C_9^2).D_6$ x 12, $C_9^2:C_6\times S_3$ x 12, $C_3^3.C_3^3.C_2^2$ x 6, $C_9\times D_9^2$ x 6, $C_9:(C_9\times S_3^2)$ x 6, $C_9^2:S_3^2$ x 6, $C_9^2:C_6^2$ x 6, $C_3^4.S_3^2$ x 6, $C_3^2:D_9^2$ x 6, $C_3^5.D_6$ x 6, $C_9.C_3^4.C_2^2$ x 5, $C_3.C_3^5.C_2^2$ x 5, $(C_3^3\times C_9).D_6$ x 5, $D_9^2:C_9$ x 5, $C_9^2.S_3^2$ x 4, $C_9^2:C_{36}$ x 2, $C_9^2:C_{36}$ x 2, $C_9^3:C_4$ x 2, $C_9:D_9^2$ x 2, $(C_3^2\times C_9):C_{36}$ x 2, $(C_3^2\times C_9):C_{36}$ x 2, $(C_3\times C_9^2):C_{12}$ x 2, $(C_3\times C_9^2):C_{12}$ x 2, $C_3\times C_9^2:C_{12}$ x 2, $(C_3^3\times C_9):C_{12}$ x 2, $C_3^4.C_6^2$ x 2, $C_3^5.C_{12}$ x 2, $C_9^2.C_{18}.C_2$

Order 2187: $C_3^2\times C_9:(C_9:C_3)$ x 2

Order 1458: $C_3\times C_9^2:C_6$ x 2

Order 729: $C_3^2\times C_9^2$ x 2

Order 243: $C_3^5$

Order 162: $C_9:D_9$ x 2

Order 81: $C_9^2$ x 2, $C_3^4$

Order 9: $C_3^2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 314928: $C_9:D_9^3.C_6$ ($C_1$)

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

Order 104976: $C_9^2\wr C_2.D_4$ ($C_3$)

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

Order 52488: $C_9^4.(C_2\times C_4)$ ($C_6$) x 2, $C_9:D_9^3$ ($C_6$)

Order 39366: $C_9^4.C_6$ ($C_2\times C_4$), $C_9^2.C_9:C_9.C_6$ ($D_4$), $C_9^2.C_3^3.C_3.C_6$ ($D_4$)

Order 26244: $C_9^4.C_2^2$ ($C_{12}$) x 2, $C_9^4.C_2^2$ ($C_2\times C_6$)

Order 19683: $C_9^4.C_3$ ($C_2^2:C_4$)

Order 13122: $C_9^2\wr C_2$ ($C_3\times D_4$) x 2, $C_9^4.C_2$ ($C_2\times C_{12}$)

Order 6561: $C_9^4$ ($C_2^2:C_{12}$)

Order 4374: $C_3\times (C_3\times C_9:(C_9:C_3)):C_2$ ($\SOPlus(4,2)$) x 2

Order 2187: $C_3^2\times C_9:(C_9:C_3)$ ($S_3^2:C_4$) x 2

Order 1458: $C_3\times C_9^2:C_6$ ($S_3^2:C_6$) x 2

Order 729: $C_3^2\times C_9^2$ ($S_3^2:C_{12}$) x 2

Order 243: $C_3^5$ ($C_3^2\wr C_2.D_4$)

Order 162: $C_9:D_9$ ($D_9^2:C_6$) x 2

Order 81: $C_9^2$ ($D_9^2:C_{12}$) x 2, $C_3^4$ ($\GL(2,3).C_2^6$)

Order 9: $C_3^2$ ($C_9^2:(S_3^2:C_{12})$) x 2

Order 1: $C_1$ ($C_9:D_9^3.C_6$)

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 $270 \times 270$ 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 $227 \times 227$ rational character table (warning: may be slow to load).