Abstract group 2239488.bp: $C_3^6.C_2^8:A_4$

• Label: 2239488.bp
• Order: \( 2^{10} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^6.C_2^8:A_4$
Order:	\(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(1146617856\)\(\medspace = 2^{19} \cdot 3^{7} \)
Composition factors:	$C_2$ x 10, $C_3$ x 7
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
Elements	1	24255	42200	25920	426024	331776	393984	995328	2239488
Conjugacy classes	1	51	34	36	1402	16	396	48	1984
Divisions	1	51	25	36	1039	8	348	24	1532
Autjugacy classes	1	17	13	8	164	2	62	4	271

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	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:	${\langle a, b, c, d, e, f, g, h, i \mid e^{12}=f^{6}=g^{6}=h^{6}=i^{2}=[b,i]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,7,2,6,9,5,4,10)(8,14,18,12,13,16,11,15,17)(19,21,25)(20,22,29)(23,26,24) \!\cdots\! \rangle$
Transitive group:	36T43683				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_3^6$ . $(C_2^8:A_4)$	$(C_3^6.C_2^7:A_4)$ . $C_2$	$(C_3^5:D_6)$ . $(C_2\wr A_4)$ (4)	$(C_3^6.C_2^5)$ . $(Q_8:A_4)$ (2)	all 40

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$2$

Subgroups

There are 153 normal subgroups (29 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 $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_2^3$
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 9 (order at least 248832) are shown, as well as all normal subgroups of any index.

Order 2239488: $C_3^6.C_2^8:A_4$

Order 1119744: $C_3^6.C_2^7:A_4$

Order 746496: $C_3^6.C_2^4.C_2^6$

Order 559872: $C_3^6.C_2\wr A_4$ x 4, $C_3^6.C_2\wr A_4$ x 4, $C_3^6.C_2\wr A_4$ x 2, $C_3^6.C_2\wr A_4$ x 2, $C_3^6.(C_2^5.\SL(2,3))$ x 2, $C_3^6.C_2\wr A_4$ x 2, $C_6\wr A_4$ x 2, $C_3^6.C_2^6.C_6.C_2$, $C_6^3.C_6^3:A_4$, $C_6^3.C_6^3:A_4$

Order 373248: $(C_3^3\times C_6^3).C_2^6$ x 13, $C_3^6.C_2^4.C_2^5$ x 12, $(C_3^2\times C_6^4).C_2^5$ x 11, $C_3^6.C_2^5.C_2^4$ x 6, $C_3^6.D_4^2.C_2^3$

Order 279936: $C_3^6.C_2^5:A_4$ x 4, $C_3^6.C_2^5:A_4$ x 4, $C_3^6.C_2^6.C_6$ x 3, $C_3^6.C_2^5.A_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.(C_2^5.A_4)$ x 2, $(C_3\times C_6^5).A_4$ x 2, $C_2^5.C_3\wr A_4$ x 2, $C_3^6.(C_2^5.A_4)$, $C_3^6.C_2^5:A_4$

Order 248832: $C_3^5.C_2^4.C_2^6$ x 2

Order 186624: $(C_3^3\times C_6^3).C_2^5$ x 8, $C_3^6.C_2^4.C_2^4$ x 5, $C_3^6.C_2^5.C_2^3$ x 3, $C_6^3.C_6^3.C_2^2$ x 2, $C_3^6.C_2^6.C_2^2$, $C_2^2\times C_3^6.C_2^6$, $(C_3\times C_6^5).C_2^3$

Order 93312: $(C_3^5\times C_6).D_4^2$ x 10, $C_3^6.C_2^5.C_2^2$ x 4, $(C_3^3\times C_6^3).C_2^4$ x 3, $C_6^3.C_6^3.C_2$ x 2, $C_2\times C_3^6.C_2^6$ x 2

Order 46656: $(C_3^3\times C_6^3).C_2^3$ x 8, $C_6^3.C_6^3$ x 4, $C_2\times C_3^6.C_2^5$ x 4, $C_3^6.C_2^4.C_2^2$ x 3, $C_6^6$, $C_3^6.C_2^5.C_2$

Order 23328: $(C_3^5\times C_6).C_4^2$ x 8, $C_2\times C_3^6.C_2^4$ x 7, $C_3^6.(C_2\times C_4^2)$ x 4, $C_3^6.C_2^5$ x 3, $C_3\times C_6^5$

Order 11664: $C_3^6.C_2^4$, $C_3^2\times C_6^4$

Order 6912: $C_6^3.C_2^5$ x 2

Order 5832: $C_3^6.C_2^3$ x 4, $C_3^3\times C_6^3$ x 3

Order 3456: $C_6^3.C_2^4$ x 2

Order 2916: $C_3^5:D_6$ x 4, $C_3^4\times C_6^2$

Order 1728: $C_6^3.Q_8$ x 4, $C_2^3\times C_6^3$ x 2, $C_6^3:C_2^3$ x 2, $C_6^3.C_2^3$ x 2

Order 1458: $C_3^5\times C_6$

Order 864: $C_6^2.D_{12}$ x 4, $C_2^2\times C_6^3$ x 2, $C_6:D_6^2$ x 2, $C_6^3.C_2^2$ x 2

Order 729: $C_3^6$

Order 432: $C_2\times C_6^3$ x 2

Order 216: $C_6^3$ x 6

Order 108: $C_3\times C_6^2$ x 2

Order 64: $C_2^6$

Order 54: $C_3^2\times C_6$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 16: $C_2^4$

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2239488: $C_3^6.C_2^8:A_4$ ($C_1$)

Order 1119744: $C_3^6.C_2^7:A_4$ ($C_2$)

Order 746496: $C_3^6.C_2^4.C_2^6$ ($C_3$)

Order 373248: $C_3^6.C_2^5.C_2^4$ ($C_6$)

Order 186624: $(C_3^3\times C_6^3).C_2^5$ ($A_4$) x 8, $C_3^6.C_2^4.C_2^4$ ($A_4$) x 5, $C_3^6.C_2^5.C_2^3$ ($A_4$) x 3, $C_6^3.C_6^3.C_2^2$ ($A_4$) x 2, $C_3^6.C_2^6.C_2^2$ ($A_4$), $C_2^2\times C_3^6.C_2^6$ ($A_4$), $(C_3\times C_6^5).C_2^3$ ($A_4$)

Order 93312: $(C_3^5\times C_6).D_4^2$ ($C_2\times A_4$) x 10, $C_3^6.C_2^5.C_2^2$ ($C_2\times A_4$) x 4, $(C_3^3\times C_6^3).C_2^4$ ($C_2\times A_4$) x 3, $C_6^3.C_6^3.C_2$ ($C_2\times A_4$) x 2, $C_2\times C_3^6.C_2^6$ ($C_2\times A_4$) x 2

Order 46656: $(C_3^3\times C_6^3).C_2^3$ ($C_2^2:A_4$) x 8, $C_6^3.C_6^3$ ($C_2^2:A_4$) x 4, $C_2\times C_3^6.C_2^5$ ($C_2^2:A_4$) x 4, $C_3^6.C_2^4.C_2^2$ ($C_2^2:A_4$) x 3, $C_6^6$ ($C_2^2:A_4$), $C_3^6.C_2^5.C_2$ ($C_2^2:A_4$)

Order 23328: $(C_3^5\times C_6).C_4^2$ ($C_2^3:A_4$) x 8, $C_2\times C_3^6.C_2^4$ ($C_2^3:A_4$) x 7, $C_3^6.(C_2\times C_4^2)$ ($C_2^3:A_4$) x 4, $C_3^6.C_2^5$ ($Q_8:A_4$) x 2, $C_3^6.C_2^5$ ($C_2^3:A_4$), $C_3\times C_6^5$ ($C_2^3:A_4$)

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

Order 6912: $C_6^3.C_2^5$ ($C_3^3:A_4$) x 2

Order 5832: $C_3^6.C_2^3$ ($C_2^5:A_4$) x 4, $C_3^3\times C_6^3$ ($C_2^5:A_4$) x 2, $C_3^3\times C_6^3$ ($C_2^5:A_4$)

Order 3456: $C_6^3.C_2^4$ ($C_2\times C_3^3:A_4$) x 2

Order 2916: $C_3^5:D_6$ ($C_2\wr A_4$) x 4, $C_3^4\times C_6^2$ ($C_2^6:A_4$)

Order 1728: $C_6^3.Q_8$ ($(C_3\times C_6^2):A_4$) x 4, $C_2^3\times C_6^3$ ($(C_3\times C_6^2):A_4$) x 2, $C_6^3:C_2^3$ ($(C_3\times C_6^2):A_4$) x 2, $C_6^3.C_2^3$ ($(C_3\times C_6^2):A_4$) x 2

Order 1458: $C_3^5\times C_6$ ($C_2^7.A_4$)

Order 864: $C_6^2.D_{12}$ ($C_6^3:A_4$) x 4, $C_2^2\times C_6^3$ ($C_6^3:A_4$) x 2, $C_6:D_6^2$ ($C_6^3:A_4$) x 2, $C_6^3.C_2^2$ ($C_6^3:A_4$) x 2

Order 729: $C_3^6$ ($C_2^8:A_4$)

Order 432: $C_2\times C_6^3$ ($(C_2\times C_6^3).A_4$) x 2

Order 216: $C_6^3$ ($C_6^3.C_2^4.C_3$) x 4, $C_6^3$ ($(C_2^2\times C_6^3).A_4$) x 2

Order 108: $C_3\times C_6^2$ ($C_6^3.C_2^4.C_6$) x 2

Order 64: $C_2^6$ ($C_3^6:(C_2^2:A_4)$)

Order 54: $C_3^2\times C_6$ ($C_3^3.C_2^6.A_4.C_2$), $C_3^2\times C_6$ ($(C_2\times C_6^3).C_2^4.C_6$)

Order 32: $C_2^5$ ($C_3^3.C_6^3:A_4$)

Order 27: $C_3^3$ ($C_2^5.C_6^3:A_4$) x 2

Order 16: $C_2^4$ ($C_3^6.C_2^4:A_4$)

Order 8: $C_2^3$ ($C_3^6.C_2^5.A_4$) x 2, $C_2^3$ ($C_3^6.C_2^5:A_4$)

Order 4: $C_2^2$ ($(C_3^5\times C_6).C_2^6.C_6$)

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

Order 1: $C_1$ ($C_3^6.C_2^8:A_4$)

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 8 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $1984 \times 1984$ character table is not available for this group.

Rational character table

The $1532 \times 1532$ rational character table is not available for this group.