Abstract group 1944.948: $C_2\times C_3^2:D_{54}$

• Label: 1944.948
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{3} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $38$
• Trans deg.: $108$
• Rank: $3$

Group information

Description:	$C_2\times C_3^2:D_{54}$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(54\)\(\medspace = 2 \cdot 3^{3} \)
Automorphism group:	$C_2\times C_3^2:C_{27}.C_9.C_2^3$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6	9	18	27	54
Elements	1	343	26	710	54	162	162	486	1944
Conjugacy classes	1	7	5	11	9	15	18	36	102
Divisions	1	7	5	11	3	5	2	4	38
Autjugacy classes	1	4	5	8	3	4	2	3	30

Dimension	1	2	4	6	12	18	36
Irr. complex chars.	8	56	26	4	8	0	0	102
Irr. rational chars.	8	8	2	8	4	4	4	38

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$108$
Rank:	$3$
Inequivalent generating triples:	$54432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	36
Arbitrary	8	8	24

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{27}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{5}, d^{a}=d^{5}, c^{b}=c^{26}d^{4}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $38$ $\langle(2,5)(3,9)(4,12)(6,14)(7,17)(8,20)(10,22)(11,23)(13,21)(15,24)(16,26)(18,27) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^2:D_{54})$
Semidirect product:	$(C_3:D_{54})$ $\,\rtimes\,$ $S_3$	$(C_6:S_3)$ $\,\rtimes\,$ $D_{27}$	$(C_3:S_3)$ $\,\rtimes\,$ $D_{54}$ (2)	$(C_3:D_{27})$ $\,\rtimes\,$ $D_6$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(S_3\times D_{27})$	$C_3$ . $(S_3\times D_{54})$	$C_9$ . $(C_6.S_3^2)$	$(C_3^2:D_6)$ . $D_9$	all 18

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 6374 subgroups in 290 conjugacy classes, 48 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:D_{54}$
Commutator:	$G' \simeq$ $C_3^2:C_{27}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3\times C_9$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2:C_{54}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_2\times C_3^2:D_{54}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times D_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2:C_{27}$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1944: $C_2\times C_3^2:D_{54}$

Order 972: $C_3^2:D_{54}$ x 4, $C_3^2:D_{54}$, $C_3^2:D_{54}$, $C_2\times C_3^2:C_{54}$

Order 648: $C_{18}:S_3^2$, $S_3\times D_{54}$

Order 486: $C_3^2:D_{27}$ x 2, $C_3^2:D_{27}$ x 2, $C_3^2:C_{54}$ x 2, $C_3^2:C_{54}$

Order 324: $S_3\times D_{27}$ x 4, $C_9:S_3^2$ x 4, $C_3\times D_{54}$ x 2, $C_3^2:D_{18}$ x 2, $C_3:D_{54}$, $S_3\times C_{54}$, $C_3:S_3\times C_{18}$

Order 243: $C_3^2:C_{27}$

Order 216: $S_3\times D_{18}$ x 2, $C_2\times D_{54}$, $C_6:S_3^2$

Order 162: $C_3\times D_{27}$ x 4, $C_3^2:D_9$ x 4, $S_3\times C_{27}$ x 2, $C_3^2:C_{18}$ x 2, $C_3\times C_{54}$ x 2, $C_3:D_{27}$ x 2, $C_3^2\times C_{18}$

Order 108: $S_3\times D_9$ x 8, $D_{54}$ x 7, $C_3:S_3^2$ x 4, $C_3\times D_{18}$ x 4, $C_3^2:D_6$ x 3, $C_3:D_{18}$ x 2, $S_3\times C_{18}$ x 2, $C_2\times C_{54}$

Order 81: $C_3\times C_{27}$ x 2, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 3, $C_2\times D_{18}$

Order 54: $C_3\times D_9$ x 8, $C_3^2:C_6$ x 6, $D_{27}$ x 6, $C_3\times C_{18}$ x 4, $C_3:D_9$ x 4, $S_3\times C_9$ x 4, $C_{54}$ x 4, $C_3^2\times C_6$

Order 36: $S_3^2$ x 12, $D_{18}$ x 8, $C_6\times S_3$ x 8, $C_6:S_3$ x 3, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 4, $C_{27}$ x 2, $C_3^3$

Order 24: $C_2\times D_6$ x 3

Order 18: $C_3\times S_3$ x 16, $D_9$ x 8, $C_3:S_3$ x 6, $C_3\times C_6$ x 5, $C_{18}$ x 5

Order 12: $D_6$ x 20, $C_2\times C_6$ x 3

Order 9: $C_3^2$ x 5, $C_9$ x 3

Order 8: $C_2^3$

Order 6: $S_3$ x 16, $C_6$ x 11

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_2\times C_3^2:D_{54}$

Order 972: $C_3^2:D_{54}$, $C_3^2:D_{54}$, $C_2\times C_3^2:C_{54}$, $C_3^2:D_{54}$

Order 648: $C_{18}:S_3^2$, $S_3\times D_{54}$

Order 486: $C_3^2:C_{54}$, $C_3^2:D_{27}$, $C_3^2:D_{27}$, $C_3^2:C_{54}$

Order 324: $C_3\times D_{54}$ x 2, $C_3^2:D_{18}$ x 2, $C_3:D_{54}$, $S_3\times C_{54}$, $S_3\times D_{27}$, $C_3:S_3\times C_{18}$, $C_9:S_3^2$

Order 243: $C_3^2:C_{27}$

Order 216: $S_3\times D_{18}$ x 2, $C_2\times D_{54}$, $C_6:S_3^2$

Order 162: $C_3\times D_{27}$ x 2, $C_3^2:D_9$ x 2, $C_3\times C_{54}$ x 2, $S_3\times C_{27}$, $C_3^2\times C_{18}$, $C_3^2:C_{18}$, $C_3:D_{27}$

Order 108: $D_{54}$ x 4, $C_3\times D_{18}$ x 4, $C_3^2:D_6$ x 3, $C_3:D_{18}$ x 2, $S_3\times C_{18}$ x 2, $S_3\times D_9$ x 2, $C_2\times C_{54}$, $C_3:S_3^2$

Order 81: $C_3\times C_{27}$ x 2, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 3, $C_2\times D_{18}$

Order 54: $C_3\times C_{18}$ x 4, $C_3\times D_9$ x 4, $C_{54}$ x 3, $C_3^2:C_6$ x 3, $D_{27}$ x 3, $C_3:D_9$ x 2, $S_3\times C_9$ x 2, $C_3^2\times C_6$

Order 36: $C_6\times S_3$ x 8, $D_{18}$ x 5, $C_6:S_3$ x 3, $S_3^2$ x 3, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 4, $C_{27}$ x 2, $C_3^3$

Order 24: $C_2\times D_6$ x 3

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 5, $C_{18}$ x 4, $D_9$ x 4, $C_3:S_3$ x 3

Order 12: $D_6$ x 11, $C_2\times C_6$ x 3

Order 9: $C_3^2$ x 5, $C_9$ x 3

Order 8: $C_2^3$

Order 6: $C_6$ x 8, $S_3$ x 8

Order 4: $C_2^2$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_2\times C_3^2:D_{54}$ ($C_1$)

Order 972: $C_3^2:D_{54}$ ($C_2$) x 4, $C_3^2:D_{54}$ ($C_2$), $C_3^2:D_{54}$ ($C_2$), $C_2\times C_3^2:C_{54}$ ($C_2$)

Order 486: $C_3^2:D_{27}$ ($C_2^2$) x 2, $C_3^2:D_{27}$ ($C_2^2$) x 2, $C_3^2:C_{54}$ ($C_2^2$) x 2, $C_3^2:C_{54}$ ($C_2^2$)

Order 324: $C_3:D_{54}$ ($S_3$), $C_3:S_3\times C_{18}$ ($S_3$)

Order 243: $C_3^2:C_{27}$ ($C_2^3$)

Order 162: $C_3^2:C_{18}$ ($D_6$) x 2, $C_3:D_{27}$ ($D_6$) x 2, $C_3^2\times C_{18}$ ($D_6$), $C_3\times C_{54}$ ($D_6$)

Order 108: $C_3^2:D_6$ ($D_9$)

Order 81: $C_3\times C_{27}$ ($C_2\times D_6$), $C_3^2\times C_9$ ($C_2\times D_6$)

Order 54: $C_3^2:C_6$ ($D_{18}$) x 2, $C_3\times C_{18}$ ($S_3^2$), $C_3^2\times C_6$ ($D_{18}$)

Order 36: $C_6:S_3$ ($D_{27}$)

Order 27: $C_3^3$ ($C_2\times D_{18}$), $C_3\times C_9$ ($S_3\times D_6$)

Order 18: $C_3:S_3$ ($D_{54}$) x 2, $C_3\times C_6$ ($D_{54}$), $C_3\times C_6$ ($S_3\times D_9$), $C_{18}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_2\times D_{54}$), $C_3^2$ ($S_3\times D_{18}$), $C_9$ ($C_6.S_3^2$)

Order 6: $C_6$ ($S_3\times D_{27}$), $C_6$ ($C_3^2:D_{18}$)

Order 3: $C_3$ ($S_3\times D_{54}$), $C_3$ ($C_2\times C_3^2:D_{18}$)

Order 2: $C_2$ ($C_3^2:D_{54}$)

Order 1: $C_1$ ($C_2\times C_3^2:D_{54}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_2\times C_3^2:D_{54}$ ($C_1$)

Order 972: $C_3^2:D_{54}$ ($C_2$), $C_3^2:D_{54}$ ($C_2$), $C_2\times C_3^2:C_{54}$ ($C_2$), $C_3^2:D_{54}$ ($C_2$)

Order 486: $C_3^2:C_{54}$ ($C_2^2$), $C_3^2:D_{27}$ ($C_2^2$), $C_3^2:D_{27}$ ($C_2^2$), $C_3^2:C_{54}$ ($C_2^2$)

Order 324: $C_3:D_{54}$ ($S_3$), $C_3:S_3\times C_{18}$ ($S_3$)

Order 243: $C_3^2:C_{27}$ ($C_2^3$)

Order 162: $C_3^2\times C_{18}$ ($D_6$), $C_3^2:C_{18}$ ($D_6$), $C_3\times C_{54}$ ($D_6$), $C_3:D_{27}$ ($D_6$)

Order 108: $C_3^2:D_6$ ($D_9$)

Order 81: $C_3\times C_{27}$ ($C_2\times D_6$), $C_3^2\times C_9$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($S_3^2$), $C_3^2\times C_6$ ($D_{18}$), $C_3^2:C_6$ ($D_{18}$)

Order 36: $C_6:S_3$ ($D_{27}$)

Order 27: $C_3^3$ ($C_2\times D_{18}$), $C_3\times C_9$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($D_{54}$), $C_3\times C_6$ ($S_3\times D_9$), $C_3:S_3$ ($D_{54}$), $C_{18}$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_2\times D_{54}$), $C_3^2$ ($S_3\times D_{18}$), $C_9$ ($C_6.S_3^2$)

Order 6: $C_6$ ($S_3\times D_{27}$), $C_6$ ($C_3^2:D_{18}$)

Order 3: $C_3$ ($S_3\times D_{54}$), $C_3$ ($C_2\times C_3^2:D_{18}$)

Order 2: $C_2$ ($C_3^2:D_{54}$)

Order 1: $C_1$ ($C_2\times C_3^2:D_{54}$)

Series

Derived series

$C_2\times C_3^2:D_{54}$

$\rhd$

$C_3^2:C_{27}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^2:D_{54}$

$\rhd$

$C_2\times C_3^2:C_{54}$

$\rhd$

$C_3^2:C_{54}$

$\rhd$

$C_3\times C_{54}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^2:D_{54}$

$\rhd$

$C_3^2:C_{27}$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.