Abstract group 648.320: $C_6\wr C_3$

• Label: 648.320
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $15$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_6\wr C_3$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3\times C_6^2:S_3^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), and metabelian.

Group statistics

Order	1	2	3	6	9	18
Elements	1	7	98	254	144	144	648
Conjugacy classes	1	3	12	64	4	4	88
Divisions	1	3	6	32	2	2	46
Autjugacy classes	1	3	4	10	1	1	20

Dimension	1	2	3	6
Irr. complex chars.	18	0	70	0	88
Irr. rational chars.	2	8	2	34	46

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	6
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{3}=b^{6}=c^{6}=d^{6}=[a,b]=[b,c]=[b,d]=[c,d]=1, c^{a}=b^{4}c^{4}d, d^{a}=b^{4}c^{3}d \rangle$
Permutation group:	Degree $15$ $\langle(10,11)(12,13)(14,15), (1,2,4)(3,5,8)(6,9,7)(10,12,14)(11,13,15), (2,5,6), (2,6,5)(4,8,9), (1,3,7)(2,5,6)(4,8,9), (12,13)(14,15), (10,11)(12,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 3 & 3 & 6 \\ 1 & 1 & 0 \\ 2 & 3 & 3 \end{array}\right), \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 5 & 5 & 2 \end{array}\right), \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{array}\right), \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 5 & 5 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrr} 0 & 0 & 6 \\ 4 & 4 & 0 \\ 2 & 0 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Transitive group:	18T188	36T968	36T1028		more information
Direct product:	$C_2$ $\, \times\, $ $(C_3^3:A_4)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $C_3$	$(C_3^2:A_4)$ $\,\rtimes\,$ $C_6$	$(C_3^2.A_4)$ $\,\rtimes\,$ $C_6$	$(C_3^2\times C_6)$ $\,\rtimes\,$ $A_4$	all 10
Trans. wreath product:	$C_6$$\ \wr\ $$C_3$
Non-split product:	$C_3^2$ . $(C_6\times A_4)$	$C_6$ . $(C_3^2:A_4)$	$C_6^2$ . $(C_3\times C_6)$	$C_3$ . $(C_6^2:C_6)$	all 8

Elements of the group are displayed as matrices in $\GL_{3}(\F_{7})$.

Homology

Abelianization:	$C_{3} \times C_{6} \simeq C_{2} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 668 subgroups in 192 conjugacy classes, 24 normal (20 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^2:A_4$
Commutator:	$G' \simeq$ $C_6^2$	$G/G' \simeq$ $C_3\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_6\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^3$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_6\wr C_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $\He_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 648: $C_6\wr C_3$

Order 324: $C_3^3:A_4$

Order 216: $C_6^2.C_6$ x 2, $C_6^3$, $C_6^2:C_6$

Order 162: $C_3^3:C_6$

Order 108: $C_3\times C_6^2$ x 3, $C_3^2.A_4$ x 2, $C_3^2:A_4$

Order 81: $C_3\wr C_3$

Order 72: $C_2\times C_6^2$ x 5, $C_2^2:C_{18}$ x 2, $C_6\times A_4$

Order 54: $C_3^2\times C_6$ x 3, $C_9:C_6$ x 2, $C_2\times \He_3$

Order 36: $C_6^2$ x 31, $C_2^2:C_9$ x 2, $C_3\times A_4$

Order 27: $C_9:C_3$ x 2, $C_3^3$, $\He_3$

Order 24: $C_2^2\times C_6$ x 5, $C_2\times A_4$

Order 18: $C_3\times C_6$ x 32, $C_{18}$ x 2

Order 12: $C_2\times C_6$ x 31, $A_4$

Order 9: $C_3^2$ x 6, $C_9$ x 2

Order 8: $C_2^3$

Order 6: $C_6$ x 32

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 6

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_6\wr C_3$

Order 324: $C_3^3:A_4$

Order 216: $C_6^3$, $C_6^2:C_6$, $C_6^2.C_6$

Order 162: $C_3^3:C_6$

Order 108: $C_3\times C_6^2$ x 3, $C_3^2:A_4$, $C_3^2.A_4$

Order 81: $C_3\wr C_3$

Order 72: $C_2\times C_6^2$ x 3, $C_6\times A_4$, $C_2^2:C_{18}$

Order 54: $C_3^2\times C_6$ x 3, $C_9:C_6$, $C_2\times \He_3$

Order 36: $C_6^2$ x 9, $C_2^2:C_9$, $C_3\times A_4$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times A_4$

Order 18: $C_3\times C_6$ x 10, $C_{18}$

Order 12: $C_2\times C_6$ x 9, $A_4$

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2^3$

Order 6: $C_6$ x 10

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_6\wr C_3$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_2$)

Order 216: $C_6^2.C_6$ ($C_3$) x 2, $C_6^3$ ($C_3$), $C_6^2:C_6$ ($C_3$)

Order 108: $C_3^2.A_4$ ($C_6$) x 2, $C_3\times C_6^2$ ($C_6$), $C_3^2:A_4$ ($C_6$)

Order 72: $C_2\times C_6^2$ ($C_3^2$)

Order 54: $C_3^2\times C_6$ ($A_4$)

Order 36: $C_6^2$ ($C_3\times C_6$)

Order 27: $C_3^3$ ($C_2\times A_4$)

Order 24: $C_2^2\times C_6$ ($\He_3$)

Order 18: $C_3\times C_6$ ($C_3\times A_4$)

Order 12: $C_2\times C_6$ ($C_2\times \He_3$)

Order 9: $C_3^2$ ($C_6\times A_4$)

Order 8: $C_2^3$ ($C_3\wr C_3$)

Order 6: $C_6$ ($C_3^2:A_4$)

Order 4: $C_2^2$ ($C_3^3:C_6$)

Order 3: $C_3$ ($C_6^2:C_6$)

Order 2: $C_2$ ($C_3^3:A_4$)

Order 1: $C_1$ ($C_6\wr C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_6\wr C_3$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_2$)

Order 216: $C_6^3$ ($C_3$), $C_6^2:C_6$ ($C_3$), $C_6^2.C_6$ ($C_3$)

Order 108: $C_3\times C_6^2$ ($C_6$), $C_3^2:A_4$ ($C_6$), $C_3^2.A_4$ ($C_6$)

Order 72: $C_2\times C_6^2$ ($C_3^2$)

Order 54: $C_3^2\times C_6$ ($A_4$)

Order 36: $C_6^2$ ($C_3\times C_6$)

Order 27: $C_3^3$ ($C_2\times A_4$)

Order 24: $C_2^2\times C_6$ ($\He_3$)

Order 18: $C_3\times C_6$ ($C_3\times A_4$)

Order 12: $C_2\times C_6$ ($C_2\times \He_3$)

Order 9: $C_3^2$ ($C_6\times A_4$)

Order 8: $C_2^3$ ($C_3\wr C_3$)

Order 6: $C_6$ ($C_3^2:A_4$)

Order 4: $C_2^2$ ($C_3^3:C_6$)

Order 3: $C_3$ ($C_6^2:C_6$)

Order 2: $C_2$ ($C_3^3:A_4$)

Order 1: $C_1$ ($C_6\wr C_3$)

Series

Derived series

$C_6\wr C_3$

$\rhd$

$C_6^2$

$\rhd$

$C_1$

Chief series

$C_6\wr C_3$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6\wr C_3$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times C_6$

$\lhd$

$C_3^2\times C_6$

Supergroups

This group is a maximal subgroup of 37 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.