Abstract group 1410048.a: $\GL(2,17).C_{18}$

• Label: 1410048.a
• Order: \( 2^{10} \cdot 3^{4} \cdot 17 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 17 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3^{2} \)
• $\card{Z(G)}$: 288
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $585$
• Trans deg.: $5184$
• Rank: $2$

Group information

Description:	$\GL(2,17).C_{18}$
Order:	\(1410048\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 17 \)
Exponent:	\(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \)
Automorphism group:	$(C_2^3\times C_{24}).\PSL(2,17).C_2$, of order \(940032\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 17 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2, $\PSL(2,17)$
Derived length:	$1$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	6	8	9	12	16	17	18	24	32	34	36	48	51	68	72	96	102	136	144	153	204	272	288	306	408	544	612	816	1224	1632	2448	4896
Elements	1	579	818	1804	3606	7280	8982	6872	29248	288	30402	21088	38912	288	46728	71552	576	576	115488	103936	576	1152	319104	1728	1152	2304	520704	1728	2304	4608	3456	4608	6912	9216	13824	27648	1410048
Conjugacy classes	1	3	5	8	15	28	39	28	104	1	117	80	144	1	180	256	2	2	432	384	2	4	1152	6	4	8	1920	6	8	16	12	16	24	32	48	96	5184
Divisions	1	3	3	5	8	9	7	9	15	1	20	13	11	1	17	19	1	1	21	15	1	1	27	1	1	1	23	1	1	1	1	1	1	1	1	1	244
Autjugacy classes	1	3	3	4	7	8	11	8	16	1	23	12	22	1	24	20	1	1	28	26	1	1	36	1	1	1	42	1	1	1	1	1	1	1	1	1	312

Minimal presentations

Permutation degree:	$585$
Transitive degree:	$5184$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Groups of Lie type:	$\GUnitary(2,17)$
Permutation group:	Degree $585$ $\langle(1,2,8,28,18,4,10,30,76,56,17,37,82,166,138,55,99,72,104,178,137,70,27,42,86,67,23,7,12,32,21,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{121} & \alpha^{215} \\ \alpha^{15} & \alpha^{274} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{250} & 0 \\ 0 & \alpha^{250} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{256} & 0 \\ 0 & \alpha^{256} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{136} & 0 \\ 0 & \alpha^{136} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{272} & 0 \\ 0 & \alpha^{272} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{212} & 0 \\ 0 & \alpha^{212} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{155} & \alpha^{175} \\ \alpha^{167} & \alpha^{13} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{125} & 0 \\ 0 & \alpha^{125} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{270} & \alpha^{279} \\ \alpha^{271} & \alpha^{23} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{192} & 0 \\ 0 & \alpha^{192} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{289}) = \GL_{2}(\F_{17}[\alpha]/(\alpha^{2} + 16 \alpha + 3))$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\GL(2,17)$ . $C_{18}$	$C_{288}$ . $\PGL(2,17)$	$(C_3\times \GL(2,17))$ . $C_6$	$(\GL(2,17):C_2)$ . $C_9$	all 58

Elements of the group are displayed as matrices in $\GUnitary(2,17)$.

Homology

Abelianization:	$C_{2} \times C_{144} \simeq C_{2} \times C_{16} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	not computed

Subgroups

There are 60 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_{288}$
Commutator:	a subgroup isomorphic to $\SL(2,17)$
Frattini:	a subgroup isomorphic to $C_{48}$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{32}:C_{16}$
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

All subgroups of index up to 17408 (order at least 81) are shown, as well as all normal subgroups of any index.

Order 1410048: $\GL(2,17).C_{18}$

Order 705024: $C_{288}.\PSL(2,17)$, $C_{144}.\PSL(2,17).C_2$, $C_{144}.\PSL(2,17).C_2$

Order 470016: $C_3\times \GL(2,17):C_2$

Order 352512: $C_{144}.\PSL(2,17)$, $C_9\times C_8.\PGL(2,17)$, $C_{72}.\PGL(2,17)$

Order 235008: $C_{96}.\PSL(2,17)$, $C_3\times C_{16}.\PGL(2,17)$, $C_3\times \GL(2,17)$

Order 176256: $C_{36}.\PGL(2,17)$, $C_{36}.\PGL(2,17)$, $C_{72}.\PSL(2,17)$

Order 156672: $\GL(2,17):C_2$

Order 117504: $\SL(2,17):C_{24}$, $C_{24}.\PGL(2,17)$, $C_{24}.\PGL(2,17)$

Order 88128: $C_{18}.\PGL(2,17)$, $\SL(2,17):C_{18}$, $\Unitary(2,17)$

Order 78336: $F_{17}\times C_{288}$, $C_{32}.\PSL(2,17)$, $C_{16}.\PGL(2,17)$, $\GL(2,17)$

Order 58752: $\SL(2,17):C_{12}$, $C_{12}.\PGL(2,17)$, $C_{12}.\PGL(2,17)$

Order 44064: $C_9\times \SL(2,17)$

Order 39168: $C_{144}\times F_{17}$, $C_{4896}:C_8$, $C_{2448}.C_{16}$, $C_8.\PGL(2,17)$, $C_8.\PGL(2,17)$, $\SL(2,17):C_8$

Order 29376: $\SL(2,17):C_6$, $\SL(2,17):C_6$, $C_6.\PGL(2,17)$

Order 26112: $C_{96}\times F_{17}$

Order 19584: $C_{136}.C_{144}$ x 2, $C_{72}\times F_{17}$ x 2, $C_{4896}:C_4$, $C_{2448}:C_8$, $C_{2448}.C_8$, $C_4.\PGL(2,17)$, $C_4.\PGL(2,17)$, $\SL(2,17):C_4$

Order 14688: $C_3\times \SL(2,17)$

Order 13056: $C_{1632}:C_8$, $C_{272}.C_{48}$, $C_{48}\times F_{17}$

Order 10368: $D_{36}.C_{144}$

Order 9792: $C_{36}\times F_{17}$ x 4, $D_{17}:C_{288}$ x 4, $D_{17}:C_{288}$ x 2, $D_{17}\times C_{288}$, $D_{17}:C_{288}$, $C_{2448}:C_4$, $\SL(2,17):C_2$, $C_{1224}.C_8$, $C_{1224}:C_8$, $\SL(2,17):C_2$, $C_2.\PGL(2,17)$

Order 9216: $D_{32}:C_{144}$

Order 8704: $C_{32}\times F_{17}$

Order 6912: $C_{18}.(C_8\times S_4).C_2$

Order 6528: $C_{136}.C_{48}$ x 2, $C_{24}\times F_{17}$ x 2, $C_{272}:C_{24}$, $C_{272}.C_{24}$, $C_{1632}:C_4$

Order 5184: $C_{288}:C_{18}$ x 2, $D_9\times C_{288}$ x 2, $C_{36}.C_{144}$, $D_{36}.C_{72}$, $C_{18}\times C_{288}$

Order 4896: $C_{18}\times F_{17}$ x 8, $C_{17}:C_{288}$ x 8, $C_{17}:C_{288}$ x 4, $C_{68}:C_{72}$ x 2, $C_{17}:C_{288}$ x 2, $D_{17}:C_{144}$ x 2, $D_{17}\times C_{144}$, $C_{17}:C_{288}$, $D_{17}:C_{144}$, $C_{1224}:C_4$, $C_{4896}$, $\SL(2,17)$

Order 4608: $D_{16}.C_{144}$ x 2, $D_{16}:C_{144}$ x 2, $C_{32}.C_{144}$, $D_{32}:C_{72}$, $C_{16}\times C_{288}$

Order 4352: $C_{272}.C_{16}$, $C_{16}\times F_{17}$, $C_{544}:C_8$

Order 3456: $C_{288}.A_4$, $C_{96}.D_{18}$, $C_{144}.S_4$, $C_{144}.S_4$, $D_{12}.C_{144}$

Order 3264: $C_{12}\times F_{17}$ x 4, $D_{17}:C_{96}$ x 4, $D_{17}:C_{96}$ x 2, $C_{136}.C_{24}$, $C_{136}:C_{24}$, $D_{17}\times C_{96}$, $C_{272}:C_{12}$, $D_{17}:C_{96}$

Order 3072: $D_{32}:C_{48}$

Order 2592: $C_{144}:C_{18}$ x 2, $C_9:C_{288}$ x 2, $D_9\times C_{144}$ x 2, $C_9\times C_{288}$ x 2, $C_{18}\times C_{144}$, $C_{36}.C_{72}$, $C_{72}.D_{18}$

Order 2448: $C_9\times F_{17}$ x 16, $C_{34}:C_{72}$ x 4, $C_{17}:C_{144}$ x 4, $C_{17}:C_{144}$ x 2, $C_{612}:C_4$, $C_{17}:C_{144}$, $D_{17}\times C_{72}$, $D_{17}:C_{72}$, $C_{2448}$

Order 2304: $D_{16}.C_{72}$ x 2, $C_{16}.C_{144}$ x 2, $D_8.C_{144}$ x 2, $D_{16}:C_{72}$ x 2, $D_8.C_{144}$ x 2, $C_8\times C_{288}$ x 2, $C_{32}.C_{72}$, $D_{32}:C_{36}$, $C_{16}\times C_{144}$, $C_{96}.S_4$

Order 2176: $C_{136}.C_{16}$ x 2, $C_8\times F_{17}$ x 2, $C_{272}.C_8$, $C_{272}:C_8$, $C_{544}:C_4$

Order 1728: $C_6\times C_{288}$ x 3, $C_{96}:C_{18}$ x 2, $S_3\times C_{288}$ x 2, $C_{288}:C_6$ x 2, $D_9\times C_{96}$ x 2, $D_{12}.C_{72}$, $C_{48}.D_{18}$, $C_{144}.A_4$, $C_{72}.S_4$, $C_{12}.C_{144}$, $C_{36}.C_{48}$, $\GL(2,3):C_{36}$

Order 1632: $C_{17}:C_{96}$ x 8, $C_6\times F_{17}$ x 8, $C_{17}:C_{96}$ x 4, $C_{204}:C_8$ x 2, $D_{17}:C_{48}$ x 2, $C_{17}:C_{96}$ x 2, $D_{17}\times C_{48}$, $C_{1632}$, $C_{136}:C_{12}$, $D_{17}:C_{48}$, $C_{17}:C_{96}$

Order 1536: $D_{16}:C_{48}$ x 2, $D_{16}.C_{48}$ x 2, $C_{16}\times C_{96}$, $D_{32}:C_{24}$, $C_{32}.C_{48}$

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

Order 1224: $C_{17}:C_{72}$ x 8, $C_{17}:C_{72}$ x 2, $C_{34}:C_{36}$ x 2, $C_{1224}$, $C_{17}:C_{72}$, $D_{17}\times C_{36}$

Order 1152: $C_4\times C_{288}$ x 4, $\OD_{64}:C_{18}$ x 2, $D_{16}:C_{36}$ x 2, $D_8.C_{72}$ x 2, $D_8:C_{72}$ x 2, $C_8\times C_{144}$ x 2, $C_8.C_{144}$ x 2, $D_{16}.C_{36}$ x 2, $\OD_{64}:C_{18}$ x 2, $C_{16}.C_{72}$ x 2, $D_{32}:C_{18}$, $D_{12}.C_{48}$, $C_{32}.D_{18}$, $C_{32}.C_{36}$, $C_{48}.S_4$, $C_{48}.S_4$, $C_{96}.A_4$, $C_{96}.A_4$

Order 1088: $D_{17}:C_{32}$ x 4, $C_4\times F_{17}$ x 4, $D_{17}:C_{32}$ x 2, $C_{32}\times D_{17}$, $C_{136}.C_8$, $C_{136}:C_8$, $D_{17}:C_{32}$, $C_{272}:C_4$

Order 1024: $D_{32}:C_{16}$

Order 864: $C_3\times C_{288}$ x 6, $C_6\times C_{144}$ x 3, $C_3:C_{288}$ x 2, $C_9:C_{96}$ x 2, $C_{48}:C_{18}$ x 2, $S_3\times C_{144}$ x 2, $C_{144}:C_6$ x 2, $D_9\times C_{48}$ x 2, $\SL(2,3):C_{36}$, $C_{12}.C_{72}$, $C_{36}.C_{24}$, $\GL(2,3):C_{18}$, $D_{12}.C_{36}$, $C_{24}.D_{18}$, $C_{72}.A_4$

Order 816: $C_3\times F_{17}$ x 16, $C_{17}:C_{48}$ x 4, $C_{102}:C_8$ x 4, $C_{17}:C_{48}$ x 2, $C_{816}$, $C_{24}\times D_{17}$, $C_{17}:C_{48}$, $C_{204}:C_4$, $D_{17}:C_{24}$

Order 768: $D_8.C_{48}$ x 2, $D_{16}:C_{24}$ x 2, $C_{16}.C_{48}$ x 2, $C_{24}.D_{16}$ x 2, $D_8.C_{48}$ x 2, $C_8\times C_{96}$ x 2, $D_{32}:C_{12}$, $C_{32}.C_{24}$, $C_{16}\times C_{48}$, $C_{32}.S_4$

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

Order 612: $C_{17}:C_{36}$ x 4, $C_9\times D_{34}$, $C_{612}$, $C_{17}:C_{36}$

Order 576: $C_2\times C_{288}$ x 13, $C_4\times C_{144}$ x 4, $C_9\times \OD_{64}$ x 3, $D_{16}:C_{18}$ x 2, $\OD_{32}:C_{18}$ x 2, $D_8:C_{36}$ x 2, $C_{288}:C_2$ x 2, $D_9\times C_{32}$ x 2, $C_9\times \SD_{64}$ x 2, $C_{16}.C_{36}$ x 2, $C_8.C_{72}$ x 2, $D_8.C_{36}$ x 2, $\OD_{32}:C_{18}$ x 2, $C_{96}:C_6$ x 2, $S_3\times C_{96}$ x 2, $C_9:\OD_{64}$, $\GL(2,3):C_{12}$, $C_{16}.D_{18}$, $D_{12}.C_{24}$, $C_{48}.A_4$, $C_{48}.A_4$, $C_{24}.S_4$, $C_9\times Q_{64}$, $C_9\times D_{32}$, $C_6\times C_{96}$, $C_8\times C_{72}$, $C_{12}.C_{48}$

Order 544: $C_2\times F_{17}$ x 8, $C_{17}:C_{32}$ x 8, $C_{17}:C_{32}$ x 4, $C_{17}:C_{32}$ x 2, $C_{68}:C_8$ x 2, $D_{17}:C_{16}$ x 2, $C_{136}:C_4$, $D_{17}:C_{16}$, $C_{16}\times D_{17}$, $C_{544}$, $C_{17}:C_{32}$

Order 512: $C_{16}\wr C_2$ x 2, $C_{32}.D_8$ x 2, $C_{16}\times C_{32}$, $D_{32}:C_8$, $C_{32}.C_{16}$

Order 432: $C_3\times C_{144}$ x 6, $C_6\times C_{72}$ x 3, $C_3:C_{144}$ x 2, $C_9:C_{48}$ x 2, $C_{24}:C_{18}$ x 2, $S_3\times C_{72}$ x 2, $C_{72}:C_6$ x 2, $D_9\times C_{24}$ x 2, $D_{12}:C_{18}$, $D_{36}:C_6$, $\SL(2,3):C_{18}$, $C_9\times \GL(2,3)$, $C_{18}.S_4$, $C_{12}.C_{36}$, $C_{36}.C_{12}$

Order 408: $C_{17}:C_{24}$ x 8, $C_{17}:C_{24}$ x 2, $C_{102}:C_4$ x 2, $C_{408}$, $C_{17}:C_{24}$, $C_{12}\times D_{17}$

Order 384: $C_4\times C_{96}$ x 4, $C_{32}.A_4$ x 2, $C_8.C_{48}$ x 2, $\OD_{64}:C_6$ x 2, $C_{12}.D_{16}$ x 2, $D_{16}:C_{12}$ x 2, $D_8.C_{24}$ x 2, $\OD_{64}:C_6$ x 2, $C_{16}.C_{24}$ x 2, $D_8:C_{24}$ x 2, $C_8\times C_{48}$ x 2, $C_{16}.S_4$, $C_{16}.S_4$, $C_{32}.C_{12}$, $D_{32}:C_6$, $C_{32}.D_6$

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

Order 306: $C_9\times D_{17}$ x 2, $C_{306}$

Order 288: $C_{288}$ x 23, $C_2\times C_{144}$ x 15, $C_9\times \OD_{32}$ x 3, $C_3\times C_{96}$ x 2, $C_3:C_{96}$ x 2, $C_9\times Q_{32}$ x 2, $C_9\times \SD_{32}$ x 2, $C_9\times D_{16}$ x 2, $C_8.C_{36}$ x 2, $D_4:C_{36}$ x 2, $C_{144}:C_2$ x 2, $C_4\times C_{72}$ x 2, $C_{16}\times D_9$ x 2, $C_{48}:C_6$ x 2, $S_3\times C_{48}$ x 2, $D_8:C_{18}$ x 2, $\OD_{16}:C_{18}$ x 2, $C_9:C_{32}$ x 2, $\GL(2,3):C_6$, $C_{24}.A_4$, $C_{24}.D_6$, $C_{24}.A_4$, $\SL(2,3):C_{12}$, $C_6\times C_{48}$, $C_{12}.C_{24}$, $C_9:\OD_{32}$, $C_8.D_{18}$

Order 272: $F_{17}$ x 16, $C_{34}:C_8$ x 4, $C_{17}:C_{16}$ x 4, $C_{17}:C_{16}$ x 2, $C_8\times D_{17}$, $C_{68}:C_4$, $D_{17}:C_8$, $C_{272}$, $C_{17}:C_{16}$

Order 256: $D_8.C_{16}$ x 2, $D_{16}:C_8$ x 2, $C_{16}.C_{16}$ x 2, $C_8.D_{16}$ x 2, $C_{16}.D_8$ x 2, $C_8\times C_{32}$ x 2, $D_{32}:C_4$, $C_{32}.C_8$, $C_{16}^2$

Order 216: $C_3\times C_{72}$ x 6, $C_6\times C_{36}$ x 3, $D_6:C_{18}$ x 2, $D_{18}:C_6$ x 2, $S_3\times C_{36}$ x 2, $C_{12}\times D_9$ x 2, $C_3:C_{72}$ x 2, $C_9:C_{24}$ x 2, $C_9\times D_{12}$, $C_3\times D_{36}$, $C_{12}.C_{18}$, $C_{36}.C_6$, $C_9\times \SL(2,3)$

Order 204: $C_{17}:C_{12}$ x 4, $C_3\times D_{34}$, $C_{204}$, $C_{17}:C_{12}$

Order 192: $C_2\times C_{96}$ x 9, $C_4\times C_{48}$ x 4, $C_3\times \OD_{64}$ x 3, $D_{16}:C_6$ x 2, $\OD_{32}:C_6$ x 2, $D_8:C_{12}$ x 2, $C_{96}:C_2$ x 2, $S_3\times C_{32}$ x 2, $C_{16}.A_4$ x 2, $C_3\times \SD_{64}$ x 2, $C_{16}.C_{12}$ x 2, $C_8.C_{24}$ x 2, $C_{12}.D_8$ x 2, $\OD_{32}:C_6$ x 2, $\GL(2,3):C_4$, $C_3:\OD_{64}$, $C_{16}.D_6$, $C_8.S_4$, $C_3\times Q_{64}$, $C_3\times D_{32}$, $C_8\times C_{24}$

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

Order 153: $C_{153}$

Order 144: $C_{144}$ x 27, $C_2\times C_{72}$ x 11, $C_9\times \OD_{16}$ x 3, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_{72}:C_2$ x 2, $D_4:C_{18}$ x 2, $C_8\times D_9$ x 2, $C_3\times C_{48}$ x 2, $C_3:C_{48}$ x 2, $C_9\times Q_{16}$ x 2, $C_9\times \SD_{16}$ x 2, $C_9\times D_8$ x 2, $C_9:C_{16}$ x 2, $C_{12}.C_{12}$, $D_{36}:C_2$, $C_{12}.A_4$, $C_4\times C_{36}$, $D_{12}:C_6$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$, $C_6\times C_{24}$, $C_9:\OD_{16}$

Order 136: $C_{17}:C_8$ x 8, $C_{17}:C_8$ x 2, $C_{34}:C_4$ x 2, $C_4\times D_{17}$, $C_{136}$, $C_{17}:C_8$

Order 128: $C_4\times C_{32}$ x 4, $\OD_{64}:C_2$ x 2, $D_{16}:C_4$ x 2, $C_{16}.D_4$ x 2, $C_8\wr C_2$ x 2, $C_8\times C_{16}$ x 2, $C_8.C_{16}$ x 2, $C_4.D_{16}$ x 2, $\OD_{64}:C_2$ x 2, $C_{16}.C_8$ x 2, $D_{32}:C_2$, $C_{32}.C_4$

Order 108: $C_3\times C_{36}$ x 6, $C_6\times C_{18}$ x 3, $C_3:C_{36}$ x 2, $C_9:C_{12}$ x 2, $S_3\times C_{18}$ x 2, $C_3\times D_{18}$ x 2

Order 102: $C_3\times D_{17}$ x 2, $C_{102}$

Order 96: $C_{96}$ x 15, $C_2\times C_{48}$ x 11, $C_3\times \OD_{32}$ x 3, $C_8.A_4$ x 2, $C_3\times Q_{32}$ x 2, $C_3\times \SD_{32}$ x 2, $C_3\times D_{16}$ x 2, $C_8.C_{12}$ x 2, $D_4:C_{12}$ x 2, $C_{48}:C_2$ x 2, $C_4\times C_{24}$ x 2, $S_3\times C_{16}$ x 2, $D_8:C_6$ x 2, $\OD_{16}:C_6$ x 2, $C_3:C_{32}$ x 2, $\Unitary(2,3)$, $\GL(2,3):C_2$, $C_3:\OD_{32}$, $C_8.D_6$

Order 81: $C_9^2$

Order 72: $C_{72}$

Order 48: $C_{48}$

Order 36: $C_{36}$

Order 32: $C_{32}$

Order 24: $C_{24}$

Order 18: $C_{18}$

Order 16: $C_{16}$

Order 12: $C_{12}$

Order 9: $C_9$

Order 8: $C_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1410048: $\GL(2,17).C_{18}$ ($C_1$)

Order 705024: $C_{288}.\PSL(2,17)$ ($C_2$), $C_{144}.\PSL(2,17).C_2$ ($C_2$), $C_{144}.\PSL(2,17).C_2$ ($C_2$)

Order 470016: $C_3\times \GL(2,17):C_2$ ($C_3$)

Order 352512: $C_{144}.\PSL(2,17)$ ($C_2^2$), $C_9\times C_8.\PGL(2,17)$ ($C_4$), $C_{72}.\PGL(2,17)$ ($C_4$)

Order 235008: $C_{96}.\PSL(2,17)$ ($C_6$), $C_3\times C_{16}.\PGL(2,17)$ ($C_6$), $C_3\times \GL(2,17)$ ($C_6$)

Order 176256: $C_{36}.\PGL(2,17)$ ($C_8$), $C_{36}.\PGL(2,17)$ ($C_8$), $C_{72}.\PSL(2,17)$ ($C_2\times C_4$)

Order 156672: $\GL(2,17):C_2$ ($C_9$)

Order 117504: $\SL(2,17):C_{24}$ ($C_2\times C_6$), $C_{24}.\PGL(2,17)$ ($C_{12}$), $C_{24}.\PGL(2,17)$ ($C_{12}$)

Order 88128: $C_{18}.\PGL(2,17)$ ($C_{16}$), $\SL(2,17):C_{18}$ ($C_2\times C_8$), $\Unitary(2,17)$ ($C_{16}$)

Order 78336: $C_{32}.\PSL(2,17)$ ($C_{18}$), $C_{16}.\PGL(2,17)$ ($C_{18}$), $\GL(2,17)$ ($C_{18}$)

Order 58752: $\SL(2,17):C_{12}$ ($C_2\times C_{12}$), $C_{12}.\PGL(2,17)$ ($C_{24}$), $C_{12}.\PGL(2,17)$ ($C_{24}$)

Order 44064: $C_9\times \SL(2,17)$ ($C_2\times C_{16}$)

Order 39168: $C_8.\PGL(2,17)$ ($C_{36}$), $C_8.\PGL(2,17)$ ($C_{36}$), $\SL(2,17):C_8$ ($C_2\times C_{18}$)

Order 29376: $\SL(2,17):C_6$ ($C_2\times C_{24}$), $\SL(2,17):C_6$ ($C_{48}$), $C_6.\PGL(2,17)$ ($C_{48}$)

Order 19584: $C_4.\PGL(2,17)$ ($C_{72}$), $C_4.\PGL(2,17)$ ($C_{72}$), $\SL(2,17):C_4$ ($C_2\times C_{36}$)

Order 14688: $C_3\times \SL(2,17)$ ($C_2\times C_{48}$)

Order 9792: $\SL(2,17):C_2$ ($C_{144}$), $\SL(2,17):C_2$ ($C_2\times C_{72}$), $C_2.\PGL(2,17)$ ($C_{144}$)

Order 4896: $\SL(2,17)$ ($C_2\times C_{144}$)

Order 288: $C_{288}$ ($\PGL(2,17)$)

Order 144: $C_{144}$ ($C_2\times \PGL(2,17)$)

Order 96: $C_{96}$ ($C_3.\PSL(2,17).C_2$)

Order 72: $C_{72}$ ($C_4\times \PSL(2,17).C_2$)

Order 48: $C_{48}$ ($C_6.\PSL(2,17).C_2$)

Order 36: $C_{36}$ ($C_8\times \PSL(2,17).C_2$)

Order 32: $C_{32}$ ($C_9.\PSL(2,17).C_2$)

Order 24: $C_{24}$ ($C_{12}.\PSL(2,17).C_2$)

Order 18: $C_{18}$ ($C_{16}\times \PGL(2,17)$)

Order 16: $C_{16}$ ($C_{18}.\PSL(2,17).C_2$)

Order 12: $C_{12}$ ($C_{24}.\PSL(2,17).C_2$)

Order 9: $C_9$ ($\GL(2,17):C_2$)

Order 8: $C_8$ ($C_{36}.\PSL(2,17).C_2$)

Order 6: $C_6$ ($C_{48}.\PSL(2,17).C_2$)

Order 4: $C_4$ ($C_{72}.\PSL(2,17).C_2$)

Order 3: $C_3$ ($C_3\times \GL(2,17):C_2$)

Order 2: $C_2$ ($C_{144}.\PSL(2,17).C_2$)

Order 1: $C_1$ ($\GL(2,17).C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $5184 \times 5184$ character table is not available for this group.

Rational character table

The $244 \times 244$ rational character table is not available for this group.