Abstract group 288.5: $C_{144}:C_2$

• Label: 288.5
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{4} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $25$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$C_{144}:C_2$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Automorphism group:	$C_2\times D_{18}:C_{12}$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	6	8	9	12	16	18	24	36	48	72	144
Elements	1	19	2	20	2	40	6	4	80	6	8	12	16	24	48	288
Conjugacy classes	1	2	1	3	1	6	3	2	8	3	4	6	8	12	24	84
Divisions	1	2	1	2	1	2	1	1	2	1	1	1	1	1	1	19
Autjugacy classes	1	2	1	2	1	2	1	1	2	1	1	1	1	1	1	19

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	16	68	0	0	0	0	0	0	0	84
Irr. rational chars.	4	4	3	2	2	1	1	1	1	19

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	48
Arbitrary	2	4	14

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{144}=1, b^{a}=b^{89} \rangle$
Permutation group:	Degree $25$ $\langle(2,6)(7,12)(8,13)(14,16)(18,20)(19,23)(21,22)(24,25), (1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 88 \end{array}\right), \left(\begin{array}{rr} 82 & 19 \\ 36 & 82 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{89})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{16}$ $\,\rtimes\,$ $D_9$	$C_{144}$ $\,\rtimes\,$ $C_2$	$C_9$ $\,\rtimes\,$ $\OD_{32}$	$(C_9:C_{16})$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $S_3$	$D_{18}$ . $C_8$	$C_{24}$ . $D_6$	$C_8$ . $D_{18}$	all 16

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 132 subgroups in 39 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $D_{18}$
Commutator:	$G' \simeq$ $C_{18}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_{24}$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{144}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{144}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $S_3\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 288: $C_{144}:C_2$

Order 144: $C_8\times D_9$, $C_{144}$, $C_9:C_{16}$

Order 96: $C_{48}:C_2$

Order 72: $C_4\times D_9$, $C_{72}$, $C_9:C_8$

Order 48: $S_3\times C_8$, $C_{48}$, $C_3:C_{16}$

Order 36: $D_{18}$, $C_{36}$, $C_9:C_4$

Order 32: $\OD_{32}$

Order 24: $C_4\times S_3$, $C_{24}$, $C_3:C_8$

Order 18: $C_{18}$, $D_9$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_9$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_{144}:C_2$

Order 144: $C_8\times D_9$, $C_{144}$, $C_9:C_{16}$

Order 96: $C_{48}:C_2$

Order 72: $C_4\times D_9$, $C_{72}$, $C_9:C_8$

Order 48: $S_3\times C_8$, $C_{48}$, $C_3:C_{16}$

Order 36: $D_{18}$, $C_{36}$, $C_9:C_4$

Order 32: $\OD_{32}$

Order 24: $C_4\times S_3$, $C_{24}$, $C_3:C_8$

Order 18: $C_{18}$, $D_9$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_9$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_{144}:C_2$ ($C_1$)

Order 144: $C_8\times D_9$ ($C_2$), $C_{144}$ ($C_2$), $C_9:C_{16}$ ($C_2$)

Order 72: $C_4\times D_9$ ($C_4$), $C_{72}$ ($C_2^2$), $C_9:C_8$ ($C_4$)

Order 48: $C_{48}$ ($S_3$)

Order 36: $D_{18}$ ($C_8$), $C_{36}$ ($C_2\times C_4$), $C_9:C_4$ ($C_8$)

Order 24: $C_{24}$ ($D_6$)

Order 18: $C_{18}$ ($C_2\times C_8$)

Order 16: $C_{16}$ ($D_9$)

Order 12: $C_{12}$ ($C_4\times S_3$)

Order 9: $C_9$ ($\OD_{32}$)

Order 8: $C_8$ ($D_{18}$)

Order 6: $C_6$ ($S_3\times C_8$)

Order 4: $C_4$ ($C_4\times D_9$)

Order 3: $C_3$ ($C_{48}:C_2$)

Order 2: $C_2$ ($C_8\times D_9$)

Order 1: $C_1$ ($C_{144}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_{144}:C_2$ ($C_1$)

Order 144: $C_8\times D_9$ ($C_2$), $C_{144}$ ($C_2$), $C_9:C_{16}$ ($C_2$)

Order 72: $C_4\times D_9$ ($C_4$), $C_{72}$ ($C_2^2$), $C_9:C_8$ ($C_4$)

Order 48: $C_{48}$ ($S_3$)

Order 36: $D_{18}$ ($C_8$), $C_{36}$ ($C_2\times C_4$), $C_9:C_4$ ($C_8$)

Order 24: $C_{24}$ ($D_6$)

Order 18: $C_{18}$ ($C_2\times C_8$)

Order 16: $C_{16}$ ($D_9$)

Order 12: $C_{12}$ ($C_4\times S_3$)

Order 9: $C_9$ ($\OD_{32}$)

Order 8: $C_8$ ($D_{18}$)

Order 6: $C_6$ ($S_3\times C_8$)

Order 4: $C_4$ ($C_4\times D_9$)

Order 3: $C_3$ ($C_{48}:C_2$)

Order 2: $C_2$ ($C_8\times D_9$)

Order 1: $C_1$ ($C_{144}:C_2$)

Series

Derived series

$C_{144}:C_2$

$\rhd$

$C_{18}$

$\rhd$

$C_1$

Chief series

$C_{144}:C_2$

$\rhd$

$C_8\times D_9$

$\rhd$

$C_{72}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{144}:C_2$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.