Abstract group 1458.311: $C_2\times C_3^3.C_3^3$

• Label: 1458.311
• Order: \( 2 \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{3} \)
• Perm deg.: $83$
• Trans deg.: $162$
• Rank: $3$

Group information

Description:	$C_2\times C_3^3.C_3^3$
Order:	\(1458\)\(\medspace = 2 \cdot 3^{6} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^3.C_3^5.C_2^2$, of order \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Composition factors:	$C_2$, $C_3$ x 6
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	6	9	18
Elements	1	1	188	188	540	540	1458
Conjugacy classes	1	1	20	20	36	36	114
Divisions	1	1	10	10	16	16	54
Autjugacy classes	1	1	8	8	7	7	32

Dimension	1	2	3	6	9	54
Irr. complex chars.	54	0	48	0	12	0	114
Irr. rational chars.	2	26	0	24	0	2	54

Minimal presentations

Permutation degree:	$83$
Transitive degree:	$162$
Rank:	$3$
Inequivalent generating triples:	$58968$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	54
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{3}=c^{3}=d^{3}=e^{9}=[a,c]=[b,c]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $83$ $\langle(6,7,8)(9,11,10)(15,16,17)(18,20,19)(24,25,26)(27,29,28)(30,33,37)(31,34,38) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^3.C_3^3)$
Semidirect product:	$(C_9.C_3^3)$ $\,\rtimes\,$ $C_6$	$(C_{18}.C_3^3)$ $\,\rtimes\,$ $C_3$	$(C_3^2.C_3^3)$ $\,\rtimes\,$ $C_6$	$(C_3^2.C_3^3)$ $\,\rtimes\,$ $C_6$ (3)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_3^4:C_3)$	$C_3$ . $(C_3^4:C_6)$	$(C_9:C_6)$ . $\He_3$	$(C_3\times C_{18})$ . $\He_3$	all 20

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:	$C_{3}^{5}$
Commutator length:	$1$

Subgroups

There are 1430 subgroups in 414 conjugacy classes, 92 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^4:C_3$
Commutator:	$G' \simeq$ $C_3^3$	$G/G' \simeq$ $C_3^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^3$	$G/\Phi \simeq$ $C_3^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_3^3.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_3^3.C_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^4:C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3.C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 1458: $C_2\times C_3^3.C_3^3$

Order 729: $C_3^3.C_3^3$

Order 486: $(C_3\times C_{18}):C_3^2$ x 5, $C_2\times C_3^2.C_3^3$ x 2, $(C_3^2\times C_{18}):C_3$ x 2, $C_{18}.C_3^3$, $(C_3\times C_{18}):C_3^2$, $C_3\times C_6.\He_3$, $(C_3\times C_{18}):C_3^2$

Order 243: $C_3^2.C_3^3$ x 5, $C_3^2.C_3^3$ x 2, $C_3^2.C_3^3$ x 2, $C_9.C_3^3$, $\He_3:C_3^2$, $C_3^2.C_3^3$, $C_3^2.C_3^3$

Order 162: $C_6.\He_3$ x 18, $C_{18}:C_3^2$ x 12, $C_6.C_3^3$ x 9, $C_6.\He_3$ x 9, $C_3^2:C_{18}$ x 6, $C_6\times \He_3$ x 3, $\He_3:C_6$ x 3, $C_3^2\times C_{18}$

Order 81: $C_3.\He_3$ x 18, $C_9:C_3^2$ x 12, $C_9.C_3^2$ x 9, $C_3.\He_3$ x 9, $C_3^2:C_9$ x 6, $\He_3:C_3$ x 3, $C_3\times \He_3$ x 3, $C_3^2\times C_9$

Order 54: $C_9:C_6$ x 47, $C_3\times C_{18}$ x 26, $C_2\times \He_3$ x 11, $C_3^2\times C_6$ x 4

Order 27: $C_9:C_3$ x 47, $C_3\times C_9$ x 26, $\He_3$ x 11, $C_3^3$ x 4

Order 18: $C_3\times C_6$ x 17, $C_{18}$ x 16

Order 9: $C_3^2$ x 17, $C_9$ x 16

Order 6: $C_6$ x 10

Order 3: $C_3$ x 10

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1458: $C_2\times C_3^3.C_3^3$

Order 729: $C_3^3.C_3^3$

Order 486: $(C_3\times C_{18}):C_3^2$ x 3, $C_{18}.C_3^3$, $C_2\times C_3^2.C_3^3$, $(C_3\times C_{18}):C_3^2$, $C_3\times C_6.\He_3$, $(C_3\times C_{18}):C_3^2$, $(C_3^2\times C_{18}):C_3$

Order 243: $C_3^2.C_3^3$ x 3, $C_9.C_3^3$, $C_3^2.C_3^3$, $\He_3:C_3^2$, $C_3^2.C_3^3$, $C_3^2.C_3^3$, $C_3^2.C_3^3$

Order 162: $C_{18}:C_3^2$ x 7, $C_6.\He_3$ x 4, $C_6.C_3^3$ x 2, $C_6\times \He_3$ x 2, $C_6.\He_3$ x 2, $C_3^2\times C_{18}$, $\He_3:C_6$, $C_3^2:C_{18}$

Order 81: $C_9:C_3^2$ x 7, $C_3.\He_3$ x 4, $C_9.C_3^2$ x 2, $C_3\times \He_3$ x 2, $C_3.\He_3$ x 2, $\He_3:C_3$, $C_3^2:C_9$, $C_3^2\times C_9$

Order 54: $C_9:C_6$ x 16, $C_3\times C_{18}$ x 10, $C_2\times \He_3$ x 5, $C_3^2\times C_6$ x 3

Order 27: $C_9:C_3$ x 16, $C_3\times C_9$ x 10, $\He_3$ x 5, $C_3^3$ x 3

Order 18: $C_3\times C_6$ x 8, $C_{18}$ x 7

Order 9: $C_3^2$ x 8, $C_9$ x 7

Order 6: $C_6$ x 6

Order 3: $C_3$ x 6

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1458: $C_2\times C_3^3.C_3^3$ ($C_1$)

Order 729: $C_3^3.C_3^3$ ($C_2$)

Order 486: $(C_3\times C_{18}):C_3^2$ ($C_3$) x 5, $C_2\times C_3^2.C_3^3$ ($C_3$) x 2, $(C_3^2\times C_{18}):C_3$ ($C_3$) x 2, $C_{18}.C_3^3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$), $C_3\times C_6.\He_3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$)

Order 243: $C_3^2.C_3^3$ ($C_6$) x 5, $C_3^2.C_3^3$ ($C_6$) x 2, $C_3^2.C_3^3$ ($C_6$) x 2, $C_9.C_3^3$ ($C_6$), $\He_3:C_3^2$ ($C_6$), $C_3^2.C_3^3$ ($C_6$), $C_3^2.C_3^3$ ($C_6$)

Order 162: $C_{18}:C_3^2$ ($C_3^2$) x 9, $C_6\times \He_3$ ($C_3^2$) x 3, $C_3^2\times C_{18}$ ($C_3^2$)

Order 81: $C_9:C_3^2$ ($C_3\times C_6$) x 9, $C_3\times \He_3$ ($C_3\times C_6$) x 3, $C_3^2\times C_9$ ($C_3\times C_6$)

Order 54: $C_3\times C_{18}$ ($\He_3$) x 6, $C_9:C_6$ ($\He_3$) x 3, $C_2\times \He_3$ ($\He_3$) x 3, $C_3^2\times C_6$ ($C_3^3$)

Order 27: $C_3\times C_9$ ($C_2\times \He_3$) x 6, $C_9:C_3$ ($C_2\times \He_3$) x 3, $\He_3$ ($C_2\times \He_3$) x 3, $C_3^3$ ($C_3^2\times C_6$)

Order 18: $C_3\times C_6$ ($C_3\times \He_3$) x 4

Order 9: $C_3^2$ ($C_6\times \He_3$) x 4

Order 6: $C_6$ ($C_3^4:C_3$)

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

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

Order 1: $C_1$ ($C_2\times C_3^3.C_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1458: $C_2\times C_3^3.C_3^3$ ($C_1$)

Order 729: $C_3^3.C_3^3$ ($C_2$)

Order 486: $(C_3\times C_{18}):C_3^2$ ($C_3$) x 3, $C_{18}.C_3^3$ ($C_3$), $C_2\times C_3^2.C_3^3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$), $C_3\times C_6.\He_3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$), $(C_3^2\times C_{18}):C_3$ ($C_3$)

Order 243: $C_3^2.C_3^3$ ($C_6$) x 3, $C_9.C_3^3$ ($C_6$), $C_3^2.C_3^3$ ($C_6$), $\He_3:C_3^2$ ($C_6$), $C_3^2.C_3^3$ ($C_6$), $C_3^2.C_3^3$ ($C_6$), $C_3^2.C_3^3$ ($C_6$)

Order 162: $C_{18}:C_3^2$ ($C_3^2$) x 6, $C_6\times \He_3$ ($C_3^2$) x 2, $C_3^2\times C_{18}$ ($C_3^2$)

Order 81: $C_9:C_3^2$ ($C_3\times C_6$) x 6, $C_3\times \He_3$ ($C_3\times C_6$) x 2, $C_3^2\times C_9$ ($C_3\times C_6$)

Order 54: $C_3\times C_{18}$ ($\He_3$), $C_3^2\times C_6$ ($C_3^3$), $C_9:C_6$ ($\He_3$), $C_2\times \He_3$ ($\He_3$)

Order 27: $C_3^3$ ($C_3^2\times C_6$), $C_9:C_3$ ($C_2\times \He_3$), $\He_3$ ($C_2\times \He_3$), $C_3\times C_9$ ($C_2\times \He_3$)

Order 18: $C_3\times C_6$ ($C_3\times \He_3$) x 3

Order 9: $C_3^2$ ($C_6\times \He_3$) x 3

Order 6: $C_6$ ($C_3^4:C_3$)

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

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

Order 1: $C_1$ ($C_2\times C_3^3.C_3^3$)

Series

Derived series

$C_2\times C_3^3.C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_9.C_3^3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^3.C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3^2\times C_6$

$\lhd$

$C_2\times C_3^3.C_3^3$

Character theory

Complex character table

See the $114 \times 114$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.