Abstract group 648.621: $C_6^3:C_3$

• Label: 648.621
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{5} \)
• Perm deg.: $18$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_6^3:C_3$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$(C_3^2\times C_6^2).S_3^3$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
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
Elements	1	7	242	398	648
Conjugacy classes	1	3	32	84	120
Divisions	1	3	16	42	62
Autjugacy classes	1	3	4	10	18

Dimension	1	2	3	6
Irr. complex chars.	54	0	66	0	120
Irr. rational chars.	2	26	2	32	62

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$1820$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{3}=b^{6}=c^{6}=d^{6}=[a,b]=[a,c]=[a,d]=[c,d]=1, c^{b}=c^{4}d, d^{b}=c^{3}d \rangle$
Permutation group:	Degree $18$ $\langle(10,11), (3,5,7)(6,9,8)(13,15,14)(16,17,18), (1,2,4)(3,5,7)(6,8,9)(16,18,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 7 & 29 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 9 \\ 9 & 34 \end{array}\right), \left(\begin{array}{rr} 19 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3^2:A_4)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $C_3$	$(C_3^2\times A_4)$ $\,\rtimes\,$ $C_6$	$(C_6\times A_4)$ $\,\rtimes\,$ $C_3^2$	$(C_3^2\times C_6)$ $\,\rtimes\,$ $A_4$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $(C_6\times A_4)$	$C_6$ . $(C_3^2\times A_4)$	$C_6^2$ . $(C_3\times C_6)$	$C_3$ . $(C_6^2:C_6)$	all 8

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

Homology

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

Subgroups

There are 1208 subgroups in 336 conjugacy classes, 84 normal (18 characteristic).

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

Special subgroups

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

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: $C_6^3:C_3$

Order 324: $C_3^3:A_4$

Order 216: $C_6^2:C_6$ x 9, $C_6^2:C_6$ x 3, $C_6^3$

Order 162: $C_6\times \He_3$

Order 108: $C_3^2:A_4$ x 9, $C_3\times C_6^2$ x 3, $C_3^2\times A_4$ x 3

Order 81: $C_3\times \He_3$

Order 72: $C_6\times A_4$ x 18, $C_2\times C_6^2$ x 7

Order 54: $C_2\times \He_3$ x 9, $C_3^2\times C_6$ x 6

Order 36: $C_6^2$ x 33, $C_3\times A_4$ x 18

Order 27: $\He_3$ x 9, $C_3^3$ x 4

Order 24: $C_2\times A_4$ x 9, $C_2^2\times C_6$ x 7

Order 18: $C_3\times C_6$ x 51

Order 12: $C_2\times C_6$ x 33, $A_4$ x 9

Order 9: $C_3^2$ x 25

Order 8: $C_2^3$

Order 6: $C_6$ x 42

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 16

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_6^3:C_3$

Order 324: $C_3^3:A_4$

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

Order 162: $C_6\times \He_3$

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

Order 81: $C_3\times \He_3$

Order 72: $C_2\times C_6^2$ x 3, $C_6\times A_4$ x 2

Order 54: $C_3^2\times C_6$ x 4, $C_2\times \He_3$

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

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

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

Order 18: $C_3\times C_6$ x 11

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

Order 9: $C_3^2$ x 5

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: $C_6^3:C_3$ ($C_1$)

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

Order 216: $C_6^2:C_6$ ($C_3$) x 9, $C_6^2:C_6$ ($C_3$) x 3, $C_6^3$ ($C_3$)

Order 108: $C_3^2:A_4$ ($C_6$) x 9, $C_3^2\times A_4$ ($C_6$) x 3, $C_3\times C_6^2$ ($C_6$)

Order 72: $C_6\times A_4$ ($C_3^2$) x 9, $C_2\times C_6^2$ ($C_3^2$) x 4

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

Order 36: $C_3\times A_4$ ($C_3\times C_6$) x 9, $C_6^2$ ($C_3\times C_6$) x 4

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

Order 24: $C_2^2\times C_6$ ($\He_3$) x 3, $C_2^2\times C_6$ ($C_3^3$)

Order 18: $C_3\times C_6$ ($C_3\times A_4$) x 4

Order 12: $C_2\times C_6$ ($C_2\times \He_3$) x 3, $C_2\times C_6$ ($C_3^2\times C_6$)

Order 9: $C_3^2$ ($C_6\times A_4$) x 4

Order 8: $C_2^3$ ($C_3\times \He_3$)

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

Order 4: $C_2^2$ ($C_6\times \He_3$)

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

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

Order 1: $C_1$ ($C_6^3:C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_6^3:C_3$ ($C_1$)

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

Order 72: $C_2\times C_6^2$ ($C_3^2$) x 2, $C_6\times A_4$ ($C_3^2$)

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

Order 36: $C_6^2$ ($C_3\times C_6$) x 2, $C_3\times A_4$ ($C_3\times C_6$)

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

Order 24: $C_2^2\times C_6$ ($C_3^3$), $C_2^2\times C_6$ ($\He_3$)

Order 18: $C_3\times C_6$ ($C_3\times A_4$) x 2

Order 12: $C_2\times C_6$ ($C_3^2\times C_6$), $C_2\times C_6$ ($C_2\times \He_3$)

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

Order 8: $C_2^3$ ($C_3\times \He_3$)

Order 6: $C_6$ ($C_3^2\times A_4$), $C_6$ ($C_3^2:A_4$)

Order 4: $C_2^2$ ($C_6\times \He_3$)

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

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

Order 1: $C_1$ ($C_6^3:C_3$)

Series

Derived series

$C_6^3:C_3$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^3:C_3$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^3:C_3$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6$

$\lhd$

$C_3^2\times C_6$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 42 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $62 \times 62$ rational character table.