Abstract group 1458.309: $(C_3\times C_{18}):\He_3$

• Label: 1458.309
• 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.: $29$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$(C_3\times C_{18}):\He_3$
Order:	\(1458\)\(\medspace = 2 \cdot 3^{6} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3^2.D_6$, 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	350	350	378	378	1458
Conjugacy classes	1	1	30	30	26	26	114
Divisions	1	1	15	15	13	13	58
Autjugacy classes	1	1	11	11	4	4	32

Dimension	1	2	3	6	9	18
Irr. complex chars.	54	0	48	0	12	0	114
Irr. rational chars.	2	26	0	24	0	6	58

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$58968$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{3}=b^{3}=c^{3}=d^{3}=e^{18}=[a,c]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $29$ $\langle(3,5,4)(9,10,11)(12,17,20)(13,15,18)(14,16,19)(21,29,24)(22,27,25)(23,28,26) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^3.C_3^3)$
Semidirect product:	$(C_3^4:C_6)$ $\,\rtimes\,$ $C_3$	$(C_3^4:C_6)$ $\,\rtimes\,$ $C_3$	$(C_3^4.C_6)$ $\,\rtimes\,$ $C_3$	$(C_3^4:C_3)$ $\,\rtimes\,$ $C_6$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(C_3\times C_6)$	$C_6$ . $(C_3^4:C_3)$	$C_3$ . $(C_3^4:C_6)$	$(C_3^3\times C_6)$ . $C_3^2$	all 14

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 2132 subgroups in 488 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:	$\operatorname{Fit} \simeq$ $(C_3\times C_{18}):\He_3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_3\times C_{18}):\He_3$	$G/R \simeq$ $C_1$
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: $(C_3\times C_{18}):\He_3$

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

Order 486: $C_2\times C_3^3:C_3^2$ x 4, $(C_3\times C_{18}):C_3^2$ x 3, $C_3^4:C_6$ x 2, $C_{18}.C_3^3$, $(C_3\times C_{18}):C_3^2$, $C_3^4.C_6$, $C_3^4:C_6$

Order 243: $C_3^3:C_3^2$ x 4, $C_3^2.C_3^3$ x 3, $C_3^4:C_3$ x 2, $C_9.C_3^3$, $\He_3:C_3^2$, $C_3^4.C_3$, $C_3^4:C_3$

Order 162: $C_3^3:C_6$ x 18, $C_{18}:C_3^2$ x 10, $C_6.C_3^3$ x 9, $C_6.\He_3$ x 9, $C_6\times \He_3$ x 8, $\He_3:C_6$ x 3, $C_3^2:C_{18}$ x 3, $C_3^3\times C_6$

Order 81: $C_3\wr C_3$ x 18, $C_9:C_3^2$ x 10, $C_3.\He_3$ x 9, $C_9.C_3^2$ x 9, $C_3\times \He_3$ x 8, $\He_3:C_3$ x 3, $C_3^2:C_9$ x 3, $C_3^4$

Order 54: $C_9:C_6$ x 43, $C_2\times \He_3$ x 24, $C_3\times C_{18}$ x 18, $C_3^2\times C_6$ x 15

Order 27: $C_9:C_3$ x 43, $\He_3$ x 24, $C_3\times C_9$ x 18, $C_3^3$ x 15

Order 18: $C_3\times C_6$ x 40, $C_{18}$ x 13

Order 9: $C_3^2$ x 40, $C_9$ x 13

Order 6: $C_6$ x 15

Order 3: $C_3$ x 15

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1458: $(C_3\times C_{18}):\He_3$

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

Order 486: $(C_3\times C_{18}):C_3^2$ x 2, $C_2\times C_3^3:C_3^2$ x 2, $C_{18}.C_3^3$, $(C_3\times C_{18}):C_3^2$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$

Order 243: $C_3^2.C_3^3$ x 2, $C_3^3:C_3^2$ x 2, $C_9.C_3^3$, $\He_3:C_3^2$, $C_3^4:C_3$, $C_3^4.C_3$, $C_3^4:C_3$

Order 162: $C_{18}:C_3^2$ x 5, $C_6\times \He_3$ x 5, $C_3^3:C_6$ x 3, $C_6.C_3^3$ x 2, $C_6.\He_3$ x 2, $C_3^3\times C_6$, $\He_3:C_6$, $C_3^2:C_{18}$

Order 81: $C_9:C_3^2$ x 5, $C_3\times \He_3$ x 5, $C_3\wr C_3$ x 3, $C_3.\He_3$ x 2, $C_9.C_3^2$ x 2, $\He_3:C_3$, $C_3^2:C_9$, $C_3^4$

Order 54: $C_9:C_6$ x 12, $C_2\times \He_3$ x 10, $C_3^2\times C_6$ x 9, $C_3\times C_{18}$ x 5

Order 27: $C_9:C_3$ x 12, $\He_3$ x 10, $C_3^3$ x 9, $C_3\times C_9$ x 5

Order 18: $C_3\times C_6$ x 18, $C_{18}$ x 4

Order 9: $C_3^2$ x 18, $C_9$ x 4

Order 6: $C_6$ x 9

Order 3: $C_3$ x 9

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1458: $(C_3\times C_{18}):\He_3$ ($C_1$)

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

Order 486: $C_2\times C_3^3:C_3^2$ ($C_3$) x 4, $(C_3\times C_{18}):C_3^2$ ($C_3$) x 3, $C_3^4:C_6$ ($C_3$) x 2, $C_{18}.C_3^3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$), $C_3^4.C_6$ ($C_3$), $C_3^4:C_6$ ($C_3$)

Order 243: $C_3^3:C_3^2$ ($C_6$) x 4, $C_3^2.C_3^3$ ($C_6$) x 3, $C_3^4:C_3$ ($C_6$) x 2, $C_9.C_3^3$ ($C_6$), $\He_3:C_3^2$ ($C_6$), $C_3^4.C_3$ ($C_6$), $C_3^4:C_3$ ($C_6$)

Order 162: $C_{18}:C_3^2$ ($C_3^2$) x 7, $C_6\times \He_3$ ($C_3^2$) x 5, $C_3^3\times C_6$ ($C_3^2$)

Order 81: $C_9:C_3^2$ ($C_3\times C_6$) x 7, $C_3\times \He_3$ ($C_3\times C_6$) x 5, $C_3^4$ ($C_3\times C_6$)

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

Order 27: $C_9:C_3$ ($C_2\times \He_3$) x 6, $C_3^3$ ($C_2\times \He_3$) x 3, $C_3\times C_9$ ($C_2\times \He_3$) 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: $C_1$ ($(C_3\times C_{18}):\He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1458: $(C_3\times C_{18}):\He_3$ ($C_1$)

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

Order 486: $(C_3\times C_{18}):C_3^2$ ($C_3$) x 2, $C_2\times C_3^3:C_3^2$ ($C_3$) x 2, $C_{18}.C_3^3$ ($C_3$), $(C_3\times C_{18}):C_3^2$ ($C_3$), $C_3^4:C_6$ ($C_3$), $C_3^4.C_6$ ($C_3$), $C_3^4:C_6$ ($C_3$)

Order 243: $C_3^2.C_3^3$ ($C_6$) x 2, $C_3^3:C_3^2$ ($C_6$) x 2, $C_9.C_3^3$ ($C_6$), $\He_3:C_3^2$ ($C_6$), $C_3^4:C_3$ ($C_6$), $C_3^4.C_3$ ($C_6$), $C_3^4:C_3$ ($C_6$)

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

Order 81: $C_9:C_3^2$ ($C_3\times C_6$) x 4, $C_3\times \He_3$ ($C_3\times C_6$) x 4, $C_3^4$ ($C_3\times C_6$)

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

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

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: $C_1$ ($(C_3\times C_{18}):\He_3$)

Series

Derived series

$(C_3\times C_{18}):\He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$(C_3\times C_{18}):\He_3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_{18}):\He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3^2\times C_6$

$\lhd$

$(C_3\times C_{18}):\He_3$

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 $58 \times 58$ rational character table.