Abstract group 10368.a: $C_{72}.D_{72}$

• Label: 10368.a
• Order: \( 2^{7} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $34$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$C_{72}.D_{72}$
Order:	\(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Exponent:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Automorphism group:	$C_{36}.C_6^2.C_2^6$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
Composition factors:	$C_2$ x 7, $C_3$ x 4
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	16	18	24	36	48	72	144
Elements	1	75	8	84	168	192	72	240	288	648	672	1296	576	4320	1728	10368
Conjugacy classes	1	3	5	8	15	28	39	52	4	117	200	444	8	1752	24	2700
Divisions	1	3	3	5	8	9	7	15	1	20	29	39	1	77	1	219
Autjugacy classes	1	3	3	4	7	6	5	10	1	11	16	16	1	26	1	111

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\GOrthPlus(2,73)$
Presentation:	$\langle a, b, c \mid a^{2}=b^{72}=c^{72}=[a,c]=[b,c]=1, b^{a}=b^{71}c^{17} \rangle$
Permutation group:	Degree $34$ $\langle(1,2,5,8,14,18,3,6,12)(4,10,11,9,16,17,15,13,7)(19,20,22,25,29,33,32,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 44 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{73})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not computed
Possibly split product:	$D_{72}$ . $C_{72}$	$C_{72}$ . $D_{72}$	$C_{36}^2$ . $D_4$	$C_{72}^2$ . $C_2$	all 152

Elements of the group are displayed as matrices in $\GL_{2}(\F_{73})$.

Homology

Abelianization:	$C_{2} \times C_{72} \simeq C_{2} \times C_{8} \times C_{9}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 3308 subgroups in 709 conjugacy classes, 156 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{72}$	$G/Z \simeq$ $D_{72}$
Commutator:	$G' \simeq$ $C_{72}$	$G/G' \simeq$ $C_2\times C_{72}$
Frattini:	$\Phi \simeq$ $C_{12}\times C_{24}$	$G/\Phi \simeq$ $C_6\times S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_{72}^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{72}.D_{72}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $D_{12}:C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 10368: $C_{72}.D_{72}$

Order 5184: $C_{72}.C_{72}$, $D_{72}:C_{36}$, $C_{72}^2$

Order 3456: $D_{24}:C_{72}$, $D_{72}:C_{24}$

Order 2592: $C_{36}\times C_{72}$ x 2, $C_{36}.C_{72}$, $C_{72}.D_{18}$, $D_{72}:C_{18}$, $C_{72}.C_{36}$, $C_{36}\wr C_2$

Order 1728: $C_{24}\times C_{72}$ x 3, $D_{24}:C_{36}$, $D_{72}:C_{12}$, $C_{24}.C_{72}$, $C_{72}.C_{24}$

Order 1296: $C_{18}\times C_{72}$ x 4, $C_{36}^2$, $C_{36}.C_{36}$, $C_9:C_{144}$, $C_9\times D_{72}$, $C_{72}:C_{18}$, $C_{72}:C_{18}$, $D_9\times C_{72}$, $C_9^2:Q_{16}$, $D_{36}:C_{18}$

Order 1152: $D_8:C_{72}$, $C_{24}\wr C_2$, $D_{72}:C_8$

Order 864: $C_{12}\times C_{72}$ x 7, $C_{24}.C_{36}$, $C_{72}.C_{12}$, $C_{12}.C_{72}$, $C_{36}.C_{24}$, $D_{12}:C_{36}$, $D_{36}:C_{12}$, $D_{24}:C_{18}$, $D_{12}.C_{36}$, $D_{72}:C_6$, $C_{24}.D_{18}$

Order 648: $C_9\times C_{72}$ x 8, $C_{18}\times C_{36}$ x 2, $C_{18}\wr C_2$, $C_9:C_{72}$, $C_9\times D_{36}$, $D_9\times C_{36}$, $C_9^2:Q_8$

Order 576: $C_8\times C_{72}$ x 7, $D_8:C_{36}$, $D_{24}:C_{12}$, $D_{72}:C_4$, $C_{72}.C_8$, $C_8.C_{72}$, $C_{24}^2$, $C_{24}.C_{24}$

Order 432: $C_6\times C_{72}$ x 14, $C_{12}\times C_{36}$ x 3, $D_{12}:C_{18}$, $D_{36}:C_6$, $C_3:C_{144}$, $C_9:C_{48}$, $C_{12}.C_{36}$, $C_{36}.C_{12}$, $C_{24}.C_{18}$, $C_9\times D_{24}$, $C_{24}:C_{18}$, $C_{24}:C_{18}$, $S_3\times C_{72}$, $C_3\times D_{72}$, $C_{72}:C_6$, $C_{72}:C_6$, $D_9\times C_{24}$, $C_{72}.C_6$

Order 384: $D_8:C_{24}$, $D_{24}:C_8$

Order 324: $C_9\times C_{36}$ x 4, $C_{18}^2$, $C_9\times D_{18}$, $C_9:C_{36}$

Order 288: $C_4\times C_{72}$ x 19, $C_{12}\times C_{24}$ x 2, $D_{24}:C_6$, $C_{24}.D_6$, $C_9\times \OD_{32}$, $C_8.C_{36}$, $D_4:C_{36}$, $C_{24}.C_{12}$, $C_{12}.C_{24}$, $C_{12}\wr C_2$, $C_{72}.C_4$, $C_9:\OD_{32}$, $D_8:C_{18}$, $\OD_{16}:C_{18}$, $D_{36}:C_4$, $D_{72}:C_2$, $C_8.D_{18}$

Order 216: $C_3\times C_{72}$ x 28, $C_6\times C_{36}$ x 7, $D_6:C_{18}$, $D_{18}:C_6$, $C_9\times D_{12}$, $S_3\times C_{36}$, $C_3\times D_{36}$, $C_{12}\times D_9$, $C_{12}.C_{18}$, $C_{36}.C_6$, $C_3:C_{72}$, $C_9:C_{24}$

Order 192: $C_8\times C_{24}$ x 3, $D_8:C_{12}$, $D_{24}:C_4$, $C_{24}.C_8$, $C_8.C_{24}$

Order 162: $C_9\times C_{18}$ x 2, $C_9\times D_9$

Order 144: $C_2\times C_{72}$ x 39, $C_4\times C_{36}$ x 7, $C_6\times C_{24}$ x 4, $C_9\times \OD_{16}$ x 2, $D_{72}$, $C_{12}.C_{12}$, $C_3^2:Q_{16}$, $C_3\times D_{24}$, $C_{24}:C_6$, $C_{24}:C_6$, $C_{72}:C_2$, $S_3\times C_{24}$, $C_{72}:C_2$, $D_4:C_{18}$, $C_8\times D_9$, $D_{36}:C_2$, $C_9:Q_{16}$, $C_3:C_{48}$, $C_9\times Q_{16}$, $C_9\times \SD_{16}$, $C_9\times D_8$, $C_{144}$, $D_{12}:C_6$, $C_{12}^2$, $C_9:\OD_{16}$, $C_9:C_{16}$

Order 128: $C_8\wr C_2$

Order 108: $C_3\times C_{36}$ x 14, $C_6\times C_{18}$ x 3, $C_3:C_{36}$, $C_9:C_{12}$, $S_3\times C_{18}$, $C_3\times D_{18}$

Order 96: $C_4\times C_{24}$ x 7, $C_3\times \OD_{32}$, $C_8.C_{12}$, $D_4:C_{12}$, $C_{24}.C_4$, $C_3:\OD_{32}$, $D_8:C_6$, $\OD_{16}:C_6$, $D_{12}:C_4$, $D_{24}:C_2$, $C_8.D_6$

Order 81: $C_9^2$

Order 72: $C_{72}$ x 77, $C_2\times C_{36}$ x 20, $C_3\times C_{24}$ x 8, $C_6\times C_{12}$ x 2, $D_4\times C_9$ x 2, $C_9:D_4$, $D_{36}$, $C_4\times D_9$, $C_9:Q_8$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3:C_{24}$, $Q_8\times C_9$, $C_9:C_8$

Order 64: $C_8.C_8$, $C_8^2$, $D_8:C_4$

Order 54: $C_3\times C_{18}$ x 7, $S_3\times C_9$, $C_3\times D_9$

Order 48: $C_2\times C_{24}$ x 15, $C_4\times C_{12}$ x 3, $C_3\times \OD_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $C_{24}:C_2$, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3\times Q_{16}$, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_{48}$, $C_3:\OD_{16}$, $C_3:C_{16}$

Order 36: $C_{36}$ x 39, $C_2\times C_{18}$ x 8, $C_3\times C_{12}$ x 4, $C_3:C_{12}$, $D_{18}$, $C_6^2$, $C_6\times S_3$, $C_9:C_4$

Order 32: $C_4\times C_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 27: $C_3\times C_9$ x 3

Order 24: $C_{24}$ x 29, $C_2\times C_{12}$ x 8, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_{18}$ x 20, $C_3\times C_6$ x 2, $C_3\times S_3$, $D_9$

Order 16: $C_2\times C_8$ x 5, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$, $C_{16}$

Order 12: $C_{12}$ x 15, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

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

Order 8: $C_8$ x 9, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

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

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 10368: $C_{72}.D_{72}$

Order 5184: $C_{72}.C_{72}$

Order 128: $C_8\wr C_2$

Order 81: $C_9^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10368: $C_{72}.D_{72}$ ($C_1$)

Order 5184: $C_{72}.C_{72}$ ($C_2$), $D_{72}:C_{36}$ ($C_2$), $C_{72}^2$ ($C_2$)

Order 3456: $D_{72}:C_{24}$ ($C_3$)

Order 2592: $C_{36}\times C_{72}$ ($C_2^2$), $D_{72}:C_{18}$ ($C_4$), $C_{72}.C_{36}$ ($C_4$)

Order 1728: $D_{72}:C_{12}$ ($C_6$), $C_{24}\times C_{72}$ ($C_6$), $C_{24}\times C_{72}$ ($S_3$), $C_{72}.C_{24}$ ($C_6$)

Order 1296: $C_{18}\times C_{72}$ ($D_4$), $C_{18}\times C_{72}$ ($C_2\times C_4$), $C_{36}^2$ ($D_4$), $C_9\times D_{72}$ ($C_8$), $C_9^2:Q_{16}$ ($C_8$)

Order 1152: $D_{72}:C_8$ ($C_9$)

Order 864: $C_{12}\times C_{72}$ ($C_2\times C_6$), $C_{12}\times C_{72}$ ($D_6$), $C_{72}.C_{12}$ ($C_{12}$), $D_{72}:C_6$ ($C_{12}$)

Order 648: $C_9\times C_{72}$ ($\SD_{16}$), $C_9\times C_{72}$ ($D_8$), $C_9\times C_{72}$ ($\OD_{16}$), $C_9\times C_{72}$ ($C_2\times C_8$), $C_{18}\times C_{36}$ ($C_2^2:C_4$)

Order 576: $D_{72}:C_4$ ($C_{18}$), $C_{72}.C_8$ ($C_{18}$), $C_{24}^2$ ($C_3\times S_3$), $C_8\times C_{72}$ ($C_{18}$), $C_8\times C_{72}$ ($D_9$)

Order 432: $C_6\times C_{72}$ ($C_2\times C_{12}$), $C_6\times C_{72}$ ($D_{12}$), $C_6\times C_{72}$ ($C_4\times S_3$), $C_6\times C_{72}$ ($C_3\times D_4$), $C_{12}\times C_{36}$ ($C_3:D_4$), $C_{12}\times C_{36}$ ($C_3\times D_4$), $C_3\times D_{72}$ ($C_{24}$), $C_{72}.C_6$ ($C_{24}$)

Order 324: $C_{18}^2$ ($C_4\wr C_2$), $C_9\times C_{36}$ ($D_4:C_4$), $C_9\times C_{36}$ ($C_2^2:C_8$)

Order 288: $C_4\times C_{72}$ ($C_2\times C_{18}$), $C_4\times C_{72}$ ($D_{18}$), $C_{12}\times C_{24}$ ($C_6\times S_3$), $C_{72}.C_4$ ($C_{36}$), $D_{72}:C_2$ ($C_{36}$)

Order 216: $C_6\times C_{36}$ ($C_2^2:C_{12}$), $C_6\times C_{36}$ ($D_6:C_4$), $C_3\times C_{72}$ ($D_{24}$), $C_3\times C_{72}$ ($C_{24}:C_2$), $C_3\times C_{72}$ ($C_{24}:C_2$), $C_3\times C_{72}$ ($S_3\times C_8$), $C_3\times C_{72}$ ($C_3\times \SD_{16}$), $C_3\times C_{72}$ ($C_3\times D_8$), $C_3\times C_{72}$ ($C_3\times \OD_{16}$), $C_3\times C_{72}$ ($C_2\times C_{24}$)

Order 192: $C_8\times C_{24}$ ($S_3\times C_9$), $C_8\times C_{24}$ ($C_3\times D_9$)

Order 162: $C_9\times C_{18}$ ($D_4:C_8$)

Order 144: $D_{72}$ ($C_{72}$), $C_9:Q_{16}$ ($C_{72}$), $C_2\times C_{72}$ ($C_2\times C_{36}$), $C_2\times C_{72}$ ($D_{36}$), $C_2\times C_{72}$ ($C_4\times D_9$), $C_2\times C_{72}$ ($D_4\times C_9$), $C_4\times C_{36}$ ($C_9:D_4$), $C_4\times C_{36}$ ($D_4\times C_9$), $C_6\times C_{24}$ ($C_3\times D_{12}$), $C_6\times C_{24}$ ($S_3\times C_{12}$), $C_{12}^2$ ($C_6\wr C_2$)

Order 108: $C_6\times C_{18}$ ($D_4:C_{12}$), $C_6\times C_{18}$ ($D_{12}:C_4$), $C_3\times C_{36}$ ($D_4:C_{12}$), $C_3\times C_{36}$ ($C_2^2:C_{24}$), $C_3\times C_{36}$ ($D_{12}:C_4$), $C_3\times C_{36}$ ($D_6:C_8$)

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

Order 81: $C_9^2$ ($C_8\wr C_2$)

Order 72: $C_2\times C_{36}$ ($C_2^2:C_{36}$), $C_2\times C_{36}$ ($D_{18}:C_4$), $C_6\times C_{12}$ ($D_6:C_{12}$), $C_{72}$ ($D_{72}$), $C_{72}$ ($C_{72}:C_2$), $C_{72}$ ($C_{72}:C_2$), $C_{72}$ ($C_8\times D_9$), $C_{72}$ ($C_9\times \SD_{16}$), $C_{72}$ ($C_9\times D_8$), $C_{72}$ ($C_9\times \OD_{16}$), $C_{72}$ ($C_2\times C_{72}$), $C_3\times C_{24}$ ($C_3\times D_{24}$), $C_3\times C_{24}$ ($C_{24}:C_6$), $C_3\times C_{24}$ ($C_{24}:C_6$), $C_3\times C_{24}$ ($S_3\times C_{24}$)

Order 64: $C_8^2$ ($C_9\times D_9$)

Order 54: $C_3\times C_{18}$ ($D_{12}:C_8$), $C_3\times C_{18}$ ($D_4:C_{24}$)

Order 48: $C_2\times C_{24}$ ($C_9\times D_{12}$), $C_2\times C_{24}$ ($S_3\times C_{36}$), $C_2\times C_{24}$ ($C_3\times D_{36}$), $C_2\times C_{24}$ ($C_{12}\times D_9$), $C_4\times C_{12}$ ($D_6:C_{18}$), $C_4\times C_{12}$ ($D_{18}:C_6$)

Order 36: $C_3\times C_{12}$ ($D_{12}:C_{12}$), $C_3\times C_{12}$ ($D_6:C_{24}$), $C_2\times C_{18}$ ($D_4:C_{36}$), $C_2\times C_{18}$ ($D_{36}:C_4$), $C_{36}$ ($D_4:C_{36}$), $C_{36}$ ($C_2^2:C_{72}$), $C_{36}$ ($D_{36}:C_4$), $C_{36}$ ($D_{18}:C_8$), $C_6^2$ ($C_{12}\wr C_2$)

Order 32: $C_4\times C_8$ ($C_9\times D_{18}$)

Order 27: $C_3\times C_9$ ($D_8:C_{24}$), $C_3\times C_9$ ($D_{24}:C_8$)

Order 24: $C_2\times C_{12}$ ($D_6:C_{36}$), $C_2\times C_{12}$ ($D_{18}:C_{12}$), $C_{24}$ ($C_9\times D_{24}$), $C_{24}$ ($C_{24}:C_{18}$), $C_{24}$ ($C_{24}:C_{18}$), $C_{24}$ ($S_3\times C_{72}$), $C_{24}$ ($C_3\times D_{72}$), $C_{24}$ ($C_{72}:C_6$), $C_{24}$ ($C_{72}:C_6$), $C_{24}$ ($D_9\times C_{24}$)

Order 18: $C_3\times C_6$ ($D_{12}:C_{24}$), $C_{18}$ ($D_{36}:C_8$), $C_{18}$ ($D_4:C_{72}$)

Order 16: $C_2\times C_8$ ($C_9\times D_{36}$), $C_2\times C_8$ ($D_9\times C_{36}$), $C_4^2$ ($C_{18}\wr C_2$)

Order 12: $C_2\times C_6$ ($D_{12}:C_{36}$), $C_2\times C_6$ ($D_{36}:C_{12}$), $C_{12}$ ($D_{12}:C_{36}$), $C_{12}$ ($D_6:C_{72}$), $C_{12}$ ($D_{36}:C_{12}$), $C_{12}$ ($D_{18}:C_{24}$)

Order 9: $C_3^2$ ($C_{24}\wr C_2$), $C_9$ ($D_8:C_{72}$), $C_9$ ($D_{72}:C_8$)

Order 8: $C_2\times C_4$ ($D_{18}:C_{36}$), $C_8$ ($C_9\times D_{72}$), $C_8$ ($C_{72}:C_{18}$), $C_8$ ($C_{72}:C_{18}$), $C_8$ ($D_9\times C_{72}$)

Order 6: $C_6$ ($D_{12}:C_{72}$), $C_6$ ($D_{36}:C_{24}$)

Order 4: $C_4$ ($(C_{18}\times C_{72}).C_2$) x 2, $C_2^2$ ($C_{36}^2.C_2$)

Order 3: $C_3$ ($D_{24}:C_{72}$), $C_3$ ($D_{72}:C_{24}$)

Order 2: $C_2$ ($C_{36}^2.C_2^2$)

Order 1: $C_1$ ($C_{72}.D_{72}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10368: $C_{72}.D_{72}$ ($C_1$)

Order 5184: $C_{72}.C_{72}$ ($C_2$)

Order 81: $C_9^2$ ($C_8\wr C_2$)

Order 1: $C_1$ ($C_{72}.D_{72}$)

Series

Derived series

$C_{72}.D_{72}$

$\rhd$

$C_{72}$

$\rhd$

$C_1$

Chief series

$C_{72}.D_{72}$

$\rhd$

$C_{72}^2$

$\rhd$

$C_{36}\times C_{72}$

$\rhd$

$C_{18}\times C_{72}$

$\rhd$

$C_9\times C_{72}$

$\rhd$

$C_3\times C_{72}$

$\rhd$

$C_{72}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{72}.D_{72}$

$\rhd$

$C_{72}$

$\rhd$

$C_{36}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_{72}$

$\lhd$

$C_2\times C_{72}$

$\lhd$

$C_4\times C_{72}$

$\lhd$

$C_8\times C_{72}$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

The $2700 \times 2700$ character table is not available for this group.

Rational character table

The $219 \times 219$ rational character table is not available for this group.