Abstract group 1728.1440: $C_9\times C_{12}.D_8$

• Label: 1728.1440
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $28$
• Trans deg.: $576$
• Rank: $2$

Group information

Description:	$C_9\times C_{12}.D_8$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_6^2.C_2^6.C_2^2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	36	72
Elements	1	3	8	60	24	64	18	192	54	224	504	576	1728
Conjugacy classes	1	3	5	7	15	8	12	38	36	40	90	96	351
Divisions	1	3	3	7	9	2	2	17	6	5	12	4	71
Autjugacy classes	1	3	3	6	9	2	2	16	6	4	11	3	66

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	72	234	45	0	0	0	0	0	0	351
Irr. rational chars.	4	10	18	4	12	8	3	9	3	71

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$576$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid b^{72}=c^{12}=[a,c]=1, a^{2}=c^{9}, b^{a}=b^{55}c^{3}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $28$ $\langle(1,2)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,15,16)(27,28), (2,5)(3,8)(6,7)(9,11,12,14,13,15,16,10) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 40 & 109 \\ 31 & 36 \end{array}\right), \left(\begin{array}{rr} 112 & 57 \\ 57 & 55 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 63 & 14 \\ 94 & 45 \end{array}\right), \left(\begin{array}{rr} 77 & 0 \\ 0 & 77 \end{array}\right), \left(\begin{array}{rr} 77 & 76 \\ 76 & 1 \end{array}\right), \left(\begin{array}{rr} 79 & 30 \\ 6 & 73 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/114\Z)$
Direct product:	$C_9$ $\, \times\, $ $(C_{12}.D_8)$
Semidirect product:	$C_3$ $\,\rtimes\,$ $(C_{36}.D_8)$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_4.D_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{36}$ . $(Q_8:S_3)$	$(C_4:C_{72})$ . $S_3$	$C_6$ . $(Q_8:C_{36})$	$(C_6\times C_{36})$ . $D_4$ (2)	all 84

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

Homology

Abelianization:	$C_{2} \times C_{36} \simeq C_{2} \times C_{4} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 640 subgroups in 230 conjugacy classes, 99 normal (93 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{18}$	$G/Z \simeq$ $D_6:C_4$
Commutator:	$G' \simeq$ $C_2\times C_{12}$	$G/G' \simeq$ $C_2\times C_{36}$
Frattini:	$\Phi \simeq$ $C_4\times C_{12}$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_{72}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_9\times C_{12}.D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2:C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 1728: $C_9\times C_{12}.D_8$

Order 864: $C_{12}:C_{72}$, $C_{12}:C_{72}$, $C_{36}.D_{12}$

Order 576: $C_{36}.D_8$, $C_{12}^2.C_2^2$

Order 432: $C_{12}:C_{36}$ x 3, $C_{36}.D_6$, $C_6\times C_{72}$, $C_{12}\times C_{36}$, $C_6:C_{72}$

Order 288: $C_4:C_{72}$ x 3, $C_{12}.D_{12}$, $C_{12}:C_{24}$, $C_{12}:C_{24}$, $C_{36}:Q_8$

Order 216: $C_6\times C_{36}$ x 3, $C_6:C_{36}$ x 2, $C_{12}.C_{18}$ x 2, $C_3\times C_{72}$, $C_3:C_{72}$

Order 192: $C_{12}.D_8$, $C_{12}.D_8$

Order 144: $C_2\times C_{72}$ x 4, $C_{12}:C_{12}$ x 3, $C_4:C_{36}$ x 3, $C_4\times C_{36}$ x 2, $C_6:C_{24}$, $Q_8\times C_{18}$, $C_{12}.D_6$, $C_6\times C_{24}$, $C_{12}^2$

Order 108: $C_3\times C_{36}$ x 5, $C_3:C_{36}$ x 2, $C_6\times C_{18}$

Order 96: $C_4:C_{24}$ x 4, $C_{12}:Q_8$, $C_{12}:Q_8$, $C_{12}:C_8$

Order 72: $C_2\times C_{36}$ x 8, $C_{72}$ x 4, $C_6\times C_{12}$ x 3, $C_6:C_{12}$ x 2, $C_3^2:Q_8$ x 2, $Q_8\times C_9$ x 2, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_4.D_8$

Order 54: $C_3\times C_{18}$ x 3

Order 48: $C_2\times C_{24}$ x 5, $C_4:C_{12}$ x 3, $C_4\times C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_6:C_8$, $C_6\times Q_8$, $C_6:Q_8$

Order 36: $C_{36}$ x 12, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 2, $C_2\times C_{18}$ x 2, $C_6^2$

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 11, $C_{24}$ x 5, $C_6:C_4$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 3, $C_2\times C_8$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 17, $C_2\times C_6$ x 3, $C_3:C_4$ x 2

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

Order 8: $C_2\times C_4$ x 5, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 9

Order 4: $C_4$ x 7, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_9\times C_{12}.D_8$

Order 864: $C_{12}:C_{72}$, $C_{12}:C_{72}$, $C_{36}.D_{12}$

Order 576: $C_{36}.D_8$, $C_{12}^2.C_2^2$

Order 432: $C_{12}:C_{36}$ x 2, $C_{36}.D_6$, $C_6\times C_{72}$, $C_{12}\times C_{36}$, $C_6:C_{72}$

Order 288: $C_4:C_{72}$ x 3, $C_{12}.D_{12}$, $C_{12}:C_{24}$, $C_{12}:C_{24}$, $C_{36}:Q_8$

Order 216: $C_6\times C_{36}$ x 3, $C_{12}.C_{18}$ x 2, $C_6:C_{36}$, $C_3\times C_{72}$, $C_3:C_{72}$

Order 192: $C_{12}.D_8$, $C_{12}.D_8$

Order 144: $C_2\times C_{72}$ x 3, $C_{12}:C_{12}$ x 2, $C_4:C_{36}$ x 2, $C_4\times C_{36}$ x 2, $C_6:C_{24}$, $Q_8\times C_{18}$, $C_{12}.D_6$, $C_6\times C_{24}$, $C_{12}^2$

Order 108: $C_3\times C_{36}$ x 5, $C_3:C_{36}$, $C_6\times C_{18}$

Order 96: $C_4:C_{24}$ x 4, $C_{12}:Q_8$, $C_{12}:Q_8$, $C_{12}:C_8$

Order 72: $C_2\times C_{36}$ x 7, $C_6\times C_{12}$ x 3, $C_{72}$ x 3, $C_3^2:Q_8$ x 2, $Q_8\times C_9$ x 2, $C_6:C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_4.D_8$

Order 54: $C_3\times C_{18}$ x 3

Order 48: $C_2\times C_{24}$ x 4, $C_4\times C_{12}$ x 3, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_6:Q_8$

Order 36: $C_{36}$ x 11, $C_3\times C_{12}$ x 5, $C_2\times C_{18}$ x 2, $C_3:C_{12}$, $C_6^2$

Order 32: $C_4:C_8$ x 2, $C_4:Q_8$

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 10, $C_{24}$ x 4, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_6:C_4$, $C_3:C_8$

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_{12}$ x 16, $C_2\times C_6$ x 3, $C_3:C_4$

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

Order 8: $C_2\times C_4$ x 4, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 9

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_9\times C_{12}.D_8$ ($C_1$)

Order 864: $C_{12}:C_{72}$ ($C_2$), $C_{12}:C_{72}$ ($C_2$), $C_{36}.D_{12}$ ($C_2$)

Order 576: $C_{12}^2.C_2^2$ ($C_3$)

Order 432: $C_{12}:C_{36}$ ($C_4$) x 2, $C_{12}\times C_{36}$ ($C_2^2$)

Order 288: $C_{12}.D_{12}$ ($C_6$), $C_4:C_{72}$ ($S_3$), $C_{12}:C_{24}$ ($C_6$), $C_{12}:C_{24}$ ($C_6$)

Order 216: $C_6\times C_{36}$ ($D_4$) x 2, $C_6\times C_{36}$ ($C_2\times C_4$)

Order 192: $C_{12}.D_8$ ($C_9$)

Order 144: $C_{12}:C_{12}$ ($C_{12}$) x 2, $C_4\times C_{36}$ ($D_6$), $C_{12}^2$ ($C_2\times C_6$)

Order 108: $C_3\times C_{36}$ ($\SD_{16}$) x 2, $C_6\times C_{18}$ ($C_2^2:C_4$), $C_3\times C_{36}$ ($Q_{16}$), $C_3\times C_{36}$ ($D_8$)

Order 96: $C_{12}:Q_8$ ($C_{18}$), $C_4:C_{24}$ ($C_3\times S_3$), $C_4:C_{24}$ ($C_{18}$), $C_{12}:C_8$ ($C_{18}$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$) x 2, $C_2\times C_{36}$ ($C_3:D_4$), $C_2\times C_{36}$ ($D_{12}$), $C_2\times C_{36}$ ($C_4\times S_3$), $C_6\times C_{12}$ ($C_2\times C_{12}$)

Order 54: $C_3\times C_{18}$ ($D_4:C_4$), $C_3\times C_{18}$ ($C_4.D_4$), $C_3\times C_{18}$ ($Q_8:C_4$)

Order 48: $C_{12}:C_4$ ($C_{36}$) x 2, $C_4\times C_{12}$ ($C_2\times C_{18}$), $C_4\times C_{12}$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$) x 2, $C_3\times C_{12}$ ($C_3\times Q_{16}$), $C_3\times C_{12}$ ($C_3\times D_8$), $C_2\times C_{18}$ ($D_6:C_4$), $C_{36}$ ($C_3:Q_{16}$), $C_{36}$ ($C_{24}:C_2$), $C_{36}$ ($Q_8:S_3$), $C_{36}$ ($C_3:D_8$), $C_6^2$ ($C_2^2:C_{12}$)

Order 32: $C_4:C_8$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($C_4.D_8$)

Order 24: $C_2\times C_{12}$ ($D_4\times C_9$) x 2, $C_2\times C_{12}$ ($C_2\times C_{36}$), $C_2\times C_{12}$ ($C_6\wr C_2$), $C_2\times C_{12}$ ($C_3\times D_{12}$), $C_2\times C_{12}$ ($S_3\times C_{12}$)

Order 18: $C_3\times C_6$ ($Q_8:C_{12}$), $C_3\times C_6$ ($D_4:C_{12}$), $C_3\times C_6$ ($C_{12}.D_4$), $C_{18}$ ($C_4.D_{12}$), $C_{18}$ ($C_6.Q_{16}$), $C_{18}$ ($D_{12}:C_4$)

Order 16: $C_4^2$ ($S_3\times C_{18}$)

Order 12: $C_{12}$ ($C_9\times \SD_{16}$) x 2, $C_2\times C_6$ ($D_6:C_{12}$), $C_2\times C_6$ ($C_2^2:C_{36}$), $C_{12}$ ($C_{12}.D_6$), $C_{12}$ ($D_{12}:C_6$), $C_{12}$ ($C_3^2:Q_{16}$), $C_{12}$ ($C_{24}:C_6$), $C_{12}$ ($C_9\times Q_{16}$), $C_{12}$ ($C_9\times D_8$)

Order 9: $C_3^2$ ($C_{12}.D_8$), $C_9$ ($C_{12}.D_8$)

Order 8: $C_2\times C_4$ ($D_6:C_{18}$), $C_2\times C_4$ ($C_9\times D_{12}$), $C_2\times C_4$ ($S_3\times C_{36}$)

Order 6: $C_6$ ($Q_8:C_{36}$), $C_6$ ($D_4:C_{36}$), $C_6$ ($C_{36}.D_4$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_6^2.D_4$), $C_6$ ($D_{12}:C_{12}$)

Order 4: $C_2^2$ ($D_6:C_{36}$), $C_4$ ($C_{36}.D_6$), $C_4$ ($D_{12}:C_{18}$), $C_4$ ($C_{24}.C_{18}$), $C_4$ ($C_{24}:C_{18}$)

Order 3: $C_3$ ($C_{36}.D_8$), $C_3$ ($C_{12}^2.C_2^2$)

Order 2: $C_2$ ($C_{36}.D_{12}$), $C_2$ ($C_9\times C_6.Q_{16}$), $C_2$ ($D_{12}:C_{36}$)

Order 1: $C_1$ ($C_9\times C_{12}.D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_9\times C_{12}.D_8$ ($C_1$)

Order 864: $C_{12}:C_{72}$ ($C_2$), $C_{12}:C_{72}$ ($C_2$), $C_{36}.D_{12}$ ($C_2$)

Order 576: $C_{12}^2.C_2^2$ ($C_3$)

Order 432: $C_{12}\times C_{36}$ ($C_2^2$), $C_{12}:C_{36}$ ($C_4$)

Order 288: $C_{12}.D_{12}$ ($C_6$), $C_4:C_{72}$ ($S_3$), $C_{12}:C_{24}$ ($C_6$), $C_{12}:C_{24}$ ($C_6$)

Order 216: $C_6\times C_{36}$ ($D_4$) x 2, $C_6\times C_{36}$ ($C_2\times C_4$)

Order 192: $C_{12}.D_8$ ($C_9$)

Order 144: $C_{12}:C_{12}$ ($C_{12}$), $C_4\times C_{36}$ ($D_6$), $C_{12}^2$ ($C_2\times C_6$)

Order 108: $C_3\times C_{36}$ ($\SD_{16}$) x 2, $C_6\times C_{18}$ ($C_2^2:C_4$), $C_3\times C_{36}$ ($Q_{16}$), $C_3\times C_{36}$ ($D_8$)

Order 96: $C_{12}:Q_8$ ($C_{18}$), $C_4:C_{24}$ ($C_3\times S_3$), $C_4:C_{24}$ ($C_{18}$), $C_{12}:C_8$ ($C_{18}$)

Order 72: $C_6\times C_{12}$ ($C_3\times D_4$) x 2, $C_2\times C_{36}$ ($C_3:D_4$), $C_2\times C_{36}$ ($D_{12}$), $C_2\times C_{36}$ ($C_4\times S_3$), $C_6\times C_{12}$ ($C_2\times C_{12}$)

Order 54: $C_3\times C_{18}$ ($D_4:C_4$), $C_3\times C_{18}$ ($C_4.D_4$), $C_3\times C_{18}$ ($Q_8:C_4$)

Order 48: $C_4\times C_{12}$ ($C_2\times C_{18}$), $C_4\times C_{12}$ ($C_6\times S_3$), $C_{12}:C_4$ ($C_{36}$)

Order 36: $C_3\times C_{12}$ ($C_3\times \SD_{16}$) x 2, $C_3\times C_{12}$ ($C_3\times Q_{16}$), $C_3\times C_{12}$ ($C_3\times D_8$), $C_2\times C_{18}$ ($D_6:C_4$), $C_{36}$ ($C_3:Q_{16}$), $C_{36}$ ($C_{24}:C_2$), $C_{36}$ ($Q_8:S_3$), $C_{36}$ ($C_3:D_8$), $C_6^2$ ($C_2^2:C_{12}$)

Order 32: $C_4:C_8$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($C_4.D_8$)

Order 24: $C_2\times C_{12}$ ($D_4\times C_9$) x 2, $C_2\times C_{12}$ ($C_2\times C_{36}$), $C_2\times C_{12}$ ($C_6\wr C_2$), $C_2\times C_{12}$ ($C_3\times D_{12}$), $C_2\times C_{12}$ ($S_3\times C_{12}$)

Order 18: $C_3\times C_6$ ($Q_8:C_{12}$), $C_3\times C_6$ ($D_4:C_{12}$), $C_3\times C_6$ ($C_{12}.D_4$), $C_{18}$ ($C_4.D_{12}$), $C_{18}$ ($C_6.Q_{16}$), $C_{18}$ ($D_{12}:C_4$)

Order 16: $C_4^2$ ($S_3\times C_{18}$)

Order 12: $C_{12}$ ($C_9\times \SD_{16}$) x 2, $C_2\times C_6$ ($D_6:C_{12}$), $C_2\times C_6$ ($C_2^2:C_{36}$), $C_{12}$ ($C_{12}.D_6$), $C_{12}$ ($D_{12}:C_6$), $C_{12}$ ($C_3^2:Q_{16}$), $C_{12}$ ($C_{24}:C_6$), $C_{12}$ ($C_9\times Q_{16}$), $C_{12}$ ($C_9\times D_8$)

Order 9: $C_3^2$ ($C_{12}.D_8$), $C_9$ ($C_{12}.D_8$)

Order 8: $C_2\times C_4$ ($D_6:C_{18}$), $C_2\times C_4$ ($C_9\times D_{12}$), $C_2\times C_4$ ($S_3\times C_{36}$)

Order 6: $C_6$ ($Q_8:C_{36}$), $C_6$ ($D_4:C_{36}$), $C_6$ ($C_{36}.D_4$), $C_6$ ($C_{12}.D_{12}$), $C_6$ ($C_6^2.D_4$), $C_6$ ($D_{12}:C_{12}$)

Order 4: $C_2^2$ ($D_6:C_{36}$), $C_4$ ($C_{36}.D_6$), $C_4$ ($D_{12}:C_{18}$), $C_4$ ($C_{24}.C_{18}$), $C_4$ ($C_{24}:C_{18}$)

Order 3: $C_3$ ($C_{36}.D_8$), $C_3$ ($C_{12}^2.C_2^2$)

Order 2: $C_2$ ($C_{36}.D_{12}$), $C_2$ ($C_9\times C_6.Q_{16}$), $C_2$ ($D_{12}:C_{36}$)

Order 1: $C_1$ ($C_9\times C_{12}.D_8$)

Series

Derived series

$C_9\times C_{12}.D_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_9\times C_{12}.D_8$

$\rhd$

$C_{12}:C_{72}$

$\rhd$

$C_{12}\times C_{36}$

$\rhd$

$C_6\times C_{36}$

$\rhd$

$C_3\times C_{36}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_9\times C_{12}.D_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{18}$

$\lhd$

$C_4\times C_{36}$

$\lhd$

$C_4:C_{72}$

Character theory

Complex character table

See the $351 \times 351$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $71 \times 71$ rational character table.