Abstract group 419904.ir: $C_3^4.D_6\wr C_3$

• Label: 419904.ir
• Order: \( 2^{6} \cdot 3^{8} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: 3
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $21$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^4.D_6\wr C_3$
Order:	\(419904\)\(\medspace = 2^{6} \cdot 3^{8} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^6:S_3^3$, of order \(10077696\)\(\medspace = 2^{9} \cdot 3^{9} \)
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	6	9	18
Elements	1	4095	9962	250326	62208	93312	419904
Conjugacy classes	1	23	77	670	18	12	801
Divisions	1	23	50	436	9	6	525
Autjugacy classes	1	9	20	85	2	1	118

Minimal presentations

Permutation degree:	$21$
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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{6}=c^{2}=d^{6}=e^{6}=f^{6}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,4)(10,12,11)(14,16)(15,17,19)(20,21), (1,3,6,2,5,8)(4,7,9)(13,15,18)(14,17,20,16,19,21), (1,2,4)(5,7)(6,9)(10,11,12)(13,14,16)(18,20,21)\rangle$
Transitive group:	36T28865				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_3^4$ . $(D_6\wr C_3)$ (2)	$S_3^3$ . $(S_3^3:C_3^2)$ (2)	$C_3^7$ . $(C_2^2\wr C_3)$	$(C_3^6.C_2^6)$ . $C_3^2$	all 41

Elements of the group are displayed as permutations of degree 21.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 108 normal subgroups (17 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 $C_3$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 32 (order at least 13122) are shown, as well as all normal subgroups of any index.

Order 419904: $C_3^4.D_6\wr C_3$

Order 209952: $S_3^3.C_3\wr A_4$ x 2, $C_3^7.C_2^3:A_4$

Order 139968: $S_3^3.S_3\wr C_3$ x 3, $C_3\times C_3^6.C_2^6$

Order 104976: $C_3^7.(C_2^2\times A_4)$ x 3, $C_3^6.(A_4\times D_6)$ x 2, $C_3^7.C_2^2:A_4$

Order 69984: $C_3\times C_3^6.C_2^5$ x 23, $C_3^3.C_6^3:A_4$ x 6, $(C_3:S_3)^3.A_4$ x 3

Order 52488: $C_3^6.(C_6\times A_4)$ x 6, $C_3^6.(C_6\times A_4)$ x 3, $C_3\times C_3^4.C_6^2.C_6$ x 2, $C_3\wr A_4.C_6$ x 2, $C_3\wr A_4.C_6$ x 2

Order 46656: $C_3\times C_3^5.C_2^6$ x 2, $C_3^6.C_2^6$

Order 34992: $C_3\times C_3^6.C_2^4$ x 205, $C_3^7.C_2^4$ x 16, $C_3^6.(C_2^2\times A_4)$ x 9, $C_3\wr A_4:C_2^2$ x 6, $C_3^6:(C_2^2:A_4)$ x 3, $C_3^6:(C_2^2\times A_4)$ x 2

Order 26244: $C_3^7.A_4$ x 3, $C_3\times C_3^4.C_6^2.C_3$ x 2, $C_3^6.C_6^2$

Order 23328: $C_3\times C_3^5.C_2^5$ x 109, $C_3^6.C_2^5$ x 26, $C_6\times C_3^5.C_2^4$ x 20, $C_3^2\times C_3^4.C_2^5$

Order 17496: $C_3\times C_3^6.C_2^3$ x 387, $C_3^7.C_2^3$ x 73, $C_3^6:(C_2\times A_4)$ x 18, $C_3^7:C_2^3$ x 14, $C_3\wr (C_2\times A_4)$ x 12, $C_3^6.(C_2\times A_4)$ x 9, $C_3\wr A_4:C_2$ x 6, $C_3\wr A_4:C_2$ x 6, $C_3\times C_3^3.C_6^2.C_6$ x 4, $C_3^5:(C_6\times A_4)$ x 2, $C_3^5:(C_6\times A_4)$ x 2, $C_3^4\times S_3^3$

Order 15552: $C_3\times C_3^4.C_2^6$ x 4, $C_3^5.C_2^6$ x 2, $C_3\times D_6\wr C_3$ x 2, $C_6\times C_3^4.C_2^5$

Order 13122: $C_3\times C_3^6.C_6$ x 3

Order 11664: $C_3^6.C_2^4$ x 3, $C_3^6:C_2^4$ x 3

Order 8748: $C_3^7:C_2^2$ x 3, $C_3^7.C_2^2$, $C_3\times C_3^6.C_2^2$, $C_3\times C_3^5.D_6$

Order 5832: $C_3^6.C_2^3$ x 6, $C_3^6:C_2^3$ x 6, $(C_3:S_3)^3$ x 3

Order 4374: $C_3\times C_3^6:C_2$ x 2, $C_3^7.C_2$

Order 2916: $C_3^5:D_6$ x 3, $C_3^6.C_2^2$ x 2, $C_3^5.D_6$

Order 2187: $C_3^7$

Order 1458: $C_3^3\wr C_2$ x 2, $C_3^5:S_3$

Order 729: $C_3^6$

Order 648: $C_3\times S_3^3$ x 2

Order 324: $C_3\wr C_2^2$ x 2

Order 216: $S_3^3$ x 2

Order 162: $C_3^3:C_6$ x 2

Order 108: $C_3:S_3^2$ x 2

Order 81: $C_3^4$ x 2

Order 54: $C_3^2:S_3$ x 2

Order 27: $C_3^3$ x 2

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 419904: $C_3^4.D_6\wr C_3$ ($C_1$)

Order 209952: $S_3^3.C_3\wr A_4$ ($C_2$) x 2, $C_3^7.C_2^3:A_4$ ($C_2$)

Order 139968: $S_3^3.S_3\wr C_3$ ($C_3$) x 3, $C_3\times C_3^6.C_2^6$ ($C_3$)

Order 104976: $C_3^7.C_2^2:A_4$ ($C_2^2$)

Order 69984: $C_3^3.C_6^3:A_4$ ($C_6$) x 6, $C_3\times C_3^6.C_2^5$ ($C_6$) x 3, $(C_3:S_3)^3.A_4$ ($C_6$) x 3

Order 46656: $C_3^6.C_2^6$ ($C_3^2$)

Order 34992: $C_3\times C_3^6.C_2^4$ ($A_4$) x 5, $C_3^6:(C_2^2:A_4)$ ($C_2\times C_6$) x 3, $C_3\times C_3^6.C_2^4$ ($C_2\times C_6$)

Order 23328: $C_3^6.C_2^5$ ($C_3\times C_6$) x 3

Order 17496: $C_3\times C_3^6.C_2^3$ ($C_2\times A_4$) x 7, $C_3^7:C_2^3$ ($C_2\times A_4$) x 6, $C_3^7.C_2^3$ ($C_2\times A_4$), $C_3^4\times S_3^3$ ($C_2\times A_4$)

Order 11664: $C_3^6:C_2^4$ ($C_3\times A_4$) x 3, $C_3^6.C_2^4$ ($C_3\times A_4$) x 2, $C_3^6.C_2^4$ ($C_6^2$)

Order 8748: $C_3^7:C_2^2$ ($C_2^2\times A_4$) x 3, $C_3^7.C_2^2$ ($C_2^2\times A_4$), $C_3\times C_3^6.C_2^2$ ($C_2^2\times A_4$), $C_3\times C_3^5.D_6$ ($C_2^2:A_4$)

Order 5832: $C_3^6.C_2^3$ ($C_6\times A_4$) x 6, $C_3^6:C_2^3$ ($C_6\times A_4$) x 6, $(C_3:S_3)^3$ ($C_6\times A_4$) x 3

Order 4374: $C_3\times C_3^6:C_2$ ($C_2^3:A_4$) x 2, $C_3^7.C_2$ ($C_2^3:A_4$)

Order 2916: $C_3^5:D_6$ ($C_2^2:C_6^2$) x 3, $C_3^6.C_2^2$ ($C_2^2:C_6^2$) x 2, $C_3^5.D_6$ ($C_2^4:C_3^2$)

Order 2187: $C_3^7$ ($C_2^2\wr C_3$)

Order 1458: $C_3^3\wr C_2$ ($C_2^5:C_3^2$) x 2, $C_3^5:S_3$ ($C_2^5:C_3^2$)

Order 729: $C_3^6$ ($C_2^4:C_6^2$)

Order 648: $C_3\times S_3^3$ ($S_3\wr C_3$) x 2

Order 324: $C_3\wr C_2^2$ ($S_3^3:C_6$) x 2

Order 216: $S_3^3$ ($S_3^3:C_3^2$) x 2

Order 162: $C_3^3:C_6$ ($S_3^3:A_4$) x 2

Order 108: $C_3:S_3^2$ ($C_6\times S_3\wr C_3$) x 2

Order 81: $C_3^4$ ($D_6\wr C_3$) x 2

Order 54: $C_3^2:S_3$ ($C_3\times S_3^3:A_4$) x 2

Order 27: $C_3^3$ ($C_3\times D_6\wr C_3$) x 2

Order 3: $C_3$ ($S_3^3.S_3\wr C_3$)

Order 1: $C_1$ ($C_3^4.D_6\wr C_3$)

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 9 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $801 \times 801$ character table is not available for this group.

Rational character table

The $525 \times 525$ rational character table is not available for this group.