Abstract group 83232.bx: $C_{408}.D_{102}$

• Label: 83232.bx
• Order: \( 2^{5} \cdot 3^{2} \cdot 17^{2} \)
• Exponent: \( 2^{3} \cdot 3 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \cdot 17 \)
• $\card{Z(G)}$: 408
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \cdot 17 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \)
• Perm deg.: $56$
• Trans deg.: $816$
• Rank: $3$

Group information

Description:	$C_{408}.D_{102}$
Order:	\(83232\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17^{2} \)
Exponent:	\(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \)
Automorphism group:	Group of order \(1671168\)\(\medspace = 2^{15} \cdot 3 \cdot 17 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_{17}$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	17	24	34	51	68	102	136	204	408
Elements	1	207	8	208	432	416	440	288	880	4128	2304	4416	13440	8832	15744	31488	83232
Conjugacy classes	1	4	5	5	17	10	22	152	44	472	1168	624	3536	1248	4704	9408	21420
Divisions	1	4	3	4	9	4	8	10	8	30	38	22	112	22	78	78	431

Minimal presentations

Permutation degree:	$56$
Transitive degree:	$816$
Rank:	$3$
Inequivalent generating triples:	not computed

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

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{102}=c^{408}=[a,c]=[b,c]=1, b^{a}=b^{101}c^{396} \rangle$
Permutation group:	Degree $56$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(20,23)(22,27)(26,30)(29,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 294 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{409})$
Direct product:	not computed
Semidirect product:	$C_{17}^2$ $\,\rtimes\,$ $(C_{24}.D_6)$	$C_3^2$ $\,\rtimes\,$ $(C_{136}.D_{34})$	$C_{51}^2$ $\,\rtimes\,$ $(\OD_{16}:C_2)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{408}$ . $D_{102}$ (2)	$D_{204}$ . $C_{204}$	$(C_{34}\times C_{408})$ . $S_3$	$C_{408}$ . $(S_3\times C_{34})$ (2)	all 168

Elements of the group are displayed as matrices in $\GL_{2}(\F_{409})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{204} \simeq C_{2}^{2} \times C_{4} \times C_{3} \times C_{17}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	not computed

Subgroups

There are 220 normal subgroups (164 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_{408}$
Commutator:	a subgroup isomorphic to $C_{102}$
Frattini:	a subgroup isomorphic to $C_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}^2$

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 5202 (order at least 16) are shown, as well as all normal subgroups of any index.

Order 83232: $C_{408}.D_{102}$

Order 41616: $C_{408}.C_{102}$ x 2, $C_{408}.C_{102}$ x 2, $C_{204}.C_{204}$, $C_{102}\times C_{408}$, $C_{204}.D_{102}$

Order 27744: $D_{68}.C_{204}$, $C_{136}.D_{102}$

Order 20808: $C_{51}:C_{408}$ x 2, $D_{51}\times C_{204}$ x 2, $C_{51}\times C_{408}$ x 2, $C_{102}\wr C_2$ x 2, $C_{51}^2:Q_8$, $C_{102}\times C_{204}$, $C_{51}\times D_{204}$

Order 13872: $C_{34}\times C_{408}$ x 3, $D_{51}\times C_{136}$ x 2, $C_{408}:C_{34}$ x 2, $C_{136}:C_{102}$ x 2, $D_{17}\times C_{408}$ x 2, $C_{68}.C_{204}$, $C_{204}.D_{34}$, $C_{204}.C_{68}$, $C_{68}.D_{102}$

Order 10404: $C_{51}\times C_{204}$ x 2, $C_{51}:C_{204}$ x 2, $C_{102}\times D_{51}$ x 2, $C_{102}^2$

Order 9248: $C_{136}.D_{34}$

Order 6936: $C_{17}\times C_{408}$ x 6, $C_{34}\times C_{204}$ x 3, $C_{51}:C_{136}$ x 2, $C_{68}\times D_{51}$ x 2, $C_{17}:C_{408}$ x 2, $D_{17}\times C_{204}$ x 2, $D_{34}:C_{102}$ x 2, $D_{102}:C_{34}$ x 2, $C_{68}.C_{102}$, $C_{51}\times D_{68}$, $C_{204}.C_{34}$, $C_{17}\times D_{204}$

Order 5202: $C_{51}\times D_{51}$ x 2, $C_{51}\times C_{102}$ x 2

Order 4896: $C_{24}.D_{102}$, $D_{12}.C_{204}$

Order 4624: $C_{136}:C_{34}$ x 2, $D_{17}\times C_{136}$ x 2, $C_{68}.C_{68}$, $D_{68}:C_{34}$, $C_{34}\times C_{136}$

Order 3468: $C_{17}\times C_{204}$ x 6, $C_{34}\times C_{102}$ x 3, $C_{17}:C_{204}$ x 2, $C_{51}\times D_{34}$ x 2, $C_{51}:C_{68}$ x 2, $C_{17}\times D_{102}$ x 2

Order 2601: $C_{51}^2$

Order 2448: $C_6\times C_{408}$ x 10, $C_{24}:C_{102}$ x 2, $S_3\times C_{408}$ x 2, $C_{408}:C_6$ x 2, $C_{24}\times D_{51}$ x 2, $C_{204}.C_{12}$, $D_{204}:C_6$, $C_{12}.C_{204}$, $D_{12}:C_{102}$

Order 2312: $D_{17}\times C_{68}$ x 2, $C_{17}:C_{136}$ x 2, $C_{17}\times C_{136}$ x 2, $C_{34}\wr C_2$ x 2, $C_{17}\times D_{68}$, $C_{34}\times C_{68}$, $C_{17}^2:Q_8$

Order 1734: $C_{17}\times C_{102}$ x 7, $C_{17}\times D_{51}$ x 2, $D_{17}\times C_{51}$ x 2

Order 1632: $\OD_{16}:C_{102}$, $C_8.D_{102}$, $D_{12}.C_{68}$, $C_{24}.D_{34}$

Order 1224: $C_3\times C_{408}$ x 20, $C_6\times C_{204}$ x 10, $S_3\times C_{204}$ x 2, $C_{51}:C_{24}$ x 2, $C_3:C_{408}$ x 2, $D_{102}:C_6$ x 2, $C_{12}\times D_{51}$ x 2, $D_6:C_{102}$ x 2, $D_{12}\times C_{51}$, $C_{12}.C_{102}$, $C_3\times D_{204}$, $C_{204}.C_6$

Order 1156: $C_{17}\times C_{68}$ x 2, $C_{17}:C_{68}$ x 2, $C_{17}\times D_{34}$ x 2, $C_{34}^2$

Order 867: $C_{17}\times C_{51}$ x 3

Order 816: $C_2\times C_{408}$ x 40, $\OD_{16}\times C_{51}$ x 3, $C_{408}:C_2$ x 2, $C_8\times D_{51}$ x 2, $C_{24}:C_{34}$ x 2, $S_3\times C_{136}$ x 2, $C_{136}:C_6$ x 2, $C_{24}\times D_{17}$ x 2, $C_{204}.C_4$, $C_{12}.C_{68}$, $C_{204}.C_4$, $D_4:C_{102}$, $D_{204}:C_2$, $D_{12}:C_{34}$, $D_{68}:C_6$

Order 612: $C_3\times C_{204}$ x 20, $C_6\times C_{102}$ x 10, $C_3\times D_{102}$ x 2, $S_3\times C_{102}$ x 2, $C_{51}:C_{12}$ x 2, $C_3:C_{204}$ x 2

Order 578: $C_{17}\times C_{34}$ x 2, $C_{17}\times D_{17}$ x 2

Order 544: $\OD_{16}:C_{34}$, $C_8.D_{34}$

Order 408: $C_{408}$ x 78, $C_2\times C_{204}$ x 40, $D_4\times C_{51}$ x 3, $C_{51}:C_8$ x 2, $C_{51}:D_4$ x 2, $C_4\times D_{51}$ x 2, $D_6:C_{34}$ x 2, $S_3\times C_{68}$ x 2, $C_{17}:C_{24}$ x 2, $C_{51}:D_4$ x 2, $C_{12}\times D_{17}$ x 2, $C_3:C_{136}$ x 2, $Q_8\times C_{51}$, $D_{204}$, $C_{51}:Q_8$, $C_{17}\times D_{12}$, $C_{51}:Q_8$, $C_3\times D_{68}$, $C_{51}:Q_8$

Order 306: $C_3\times C_{102}$ x 28, $C_3\times D_{51}$ x 2, $S_3\times C_{51}$ x 2

Order 289: $C_{17}^2$

Order 288: $C_{24}.D_6$

Order 272: $C_2\times C_{136}$ x 12, $\OD_{16}\times C_{17}$ x 3, $C_{136}:C_2$ x 2, $C_8\times D_{17}$ x 2, $D_4:C_{34}$, $D_{68}:C_2$, $C_{17}:\OD_{16}$

Order 204: $C_{204}$ x 78, $C_2\times C_{102}$ x 40, $C_3\times D_{34}$ x 2, $C_{51}:C_4$ x 2, $C_{17}:C_{12}$ x 2, $D_{102}$ x 2, $S_3\times C_{34}$ x 2, $C_3:C_{68}$ x 2

Order 153: $C_3\times C_{51}$ x 10

Order 144: $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_{12}.C_{12}$, $D_{12}:C_6$, $C_6\times C_{24}$

Order 136: $C_{136}$ x 22, $C_2\times C_{68}$ x 12, $D_4\times C_{17}$ x 3, $C_{17}:D_4$ x 2, $C_4\times D_{17}$ x 2, $C_{17}:C_8$ x 2, $D_{68}$, $C_{17}:Q_8$, $Q_8\times C_{17}$

Order 102: $C_{102}$ x 112, $D_{51}$ x 2, $C_3\times D_{17}$ x 2, $S_3\times C_{17}$ x 2

Order 96: $\OD_{16}:C_6$, $C_8.D_6$

Order 72: $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_3\times C_{24}$ x 2, $C_3:C_{24}$ x 2, $C_6\times C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 68: $C_{68}$ x 22, $C_2\times C_{34}$ x 12, $D_{34}$ x 2, $C_{17}:C_4$ x 2

Order 51: $C_{51}$ x 38

Order 48: $C_2\times C_{24}$ x 5, $C_3\times \OD_{16}$ x 3, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $D_4:C_6$, $D_{12}:C_2$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 34: $C_{34}$ x 30, $D_{17}$ x 2

Order 32: $\OD_{16}:C_2$

Order 24: $C_{24}$ x 8, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_3:C_8$ x 2, $D_{12}$, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 17: $C_{17}$ x 10

Order 16: $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2

Order 9: $C_3^2$

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 83232: $C_{408}.D_{102}$ ($C_1$)

Order 41616: $C_{408}.C_{102}$ ($C_2$) x 2, $C_{408}.C_{102}$ ($C_2$) x 2, $C_{204}.C_{204}$ ($C_2$), $C_{102}\times C_{408}$ ($C_2$), $C_{204}.D_{102}$ ($C_2$)

Order 27744: $C_{136}.D_{102}$ ($C_3$)

Order 20808: $C_{51}:C_{408}$ ($C_2^2$) x 2, $D_{51}\times C_{204}$ ($C_2^2$) x 2, $C_{51}\times C_{408}$ ($C_2^2$) x 2, $C_{102}\wr C_2$ ($C_4$) x 2, $C_{51}^2:Q_8$ ($C_4$), $C_{102}\times C_{204}$ ($C_2^2$), $C_{51}\times D_{204}$ ($C_4$)

Order 13872: $D_{51}\times C_{136}$ ($C_6$) x 2, $C_{408}:C_{34}$ ($C_6$) x 2, $C_{204}.C_{68}$ ($C_6$), $C_{68}.D_{102}$ ($C_6$), $C_{34}\times C_{408}$ ($C_6$), $C_{34}\times C_{408}$ ($S_3$)

Order 10404: $C_{51}:C_{204}$ ($C_2\times C_4$) x 2, $C_{102}\times D_{51}$ ($C_2\times C_4$) x 2, $C_{51}\times C_{204}$ ($C_2^3$), $C_{51}\times C_{204}$ ($C_2\times C_4$), $C_{102}^2$ ($C_2\times C_4$)

Order 6936: $C_{51}:C_{136}$ ($C_2\times C_6$) x 2, $C_{68}\times D_{51}$ ($C_2\times C_6$) x 2, $C_{17}\times C_{408}$ ($C_2\times C_6$) x 2, $C_{17}\times C_{408}$ ($D_6$) x 2, $D_{102}:C_{34}$ ($C_{12}$) x 2, $C_{34}\times C_{204}$ ($C_2\times C_6$), $C_{34}\times C_{204}$ ($D_6$), $C_{204}.C_{34}$ ($C_{12}$), $C_{17}\times D_{204}$ ($C_{12}$)

Order 5202: $C_{51}\times C_{102}$ ($C_2^2\times C_4$)

Order 4896: $C_{24}.D_{102}$ ($C_{17}$)

Order 4624: $C_{34}\times C_{136}$ ($C_3\times S_3$)

Order 3468: $C_{51}:C_{68}$ ($C_2\times C_{12}$) x 2, $C_{17}\times D_{102}$ ($C_2\times C_{12}$) x 2, $C_{17}\times C_{204}$ ($C_2\times C_{12}$), $C_{17}\times C_{204}$ ($C_4\times S_3$), $C_{17}\times C_{204}$ ($C_2^2\times C_6$), $C_{17}\times C_{204}$ ($C_2\times D_6$), $C_{34}\times C_{102}$ ($C_2\times C_{12}$), $C_{34}\times C_{102}$ ($C_4\times S_3$)

Order 2601: $C_{51}^2$ ($\OD_{16}:C_2$)

Order 2448: $C_{408}:C_6$ ($C_{34}$) x 2, $C_{24}\times D_{51}$ ($C_{34}$) x 2, $C_{204}.C_{12}$ ($C_{34}$), $D_{204}:C_6$ ($C_{34}$), $C_6\times C_{408}$ ($C_{34}$), $C_6\times C_{408}$ ($D_{17}$)

Order 2312: $C_{17}\times C_{136}$ ($C_6\times S_3$) x 2, $C_{34}\times C_{68}$ ($C_6\times S_3$)

Order 1734: $C_{17}\times C_{102}$ ($C_2^2\times C_{12}$), $C_{17}\times C_{102}$ ($C_4\times D_6$)

Order 1632: $C_8.D_{102}$ ($C_{51}$)

Order 1224: $C_3\times C_{408}$ ($C_2\times C_{34}$) x 2, $C_3\times C_{408}$ ($D_{34}$) x 2, $C_{51}:C_{24}$ ($C_2\times C_{34}$) x 2, $D_{102}:C_6$ ($C_{68}$) x 2, $C_{12}\times D_{51}$ ($C_2\times C_{34}$) x 2, $C_6\times C_{204}$ ($C_2\times C_{34}$), $C_6\times C_{204}$ ($D_{34}$), $C_3\times D_{204}$ ($C_{68}$), $C_{204}.C_6$ ($C_{68}$)

Order 1156: $C_{17}\times C_{68}$ ($C_6\times D_6$), $C_{17}\times C_{68}$ ($S_3\times C_{12}$), $C_{34}^2$ ($S_3\times C_{12}$)

Order 867: $C_{17}\times C_{51}$ ($\OD_{16}:C_6$), $C_{17}\times C_{51}$ ($C_8.D_6$)

Order 816: $C_{408}:C_2$ ($C_{102}$) x 2, $C_8\times D_{51}$ ($C_{102}$) x 2, $C_2\times C_{408}$ ($C_{102}$), $C_2\times C_{408}$ ($D_{51}$), $C_2\times C_{408}$ ($C_3\times D_{17}$), $C_2\times C_{408}$ ($S_3\times C_{17}$), $C_{204}.C_4$ ($C_{102}$), $D_{204}:C_2$ ($C_{102}$)

Order 612: $C_3\times D_{102}$ ($C_2\times C_{68}$) x 2, $C_{51}:C_{12}$ ($C_2\times C_{68}$) x 2, $C_6\times C_{102}$ ($C_2\times C_{68}$), $C_6\times C_{102}$ ($C_4\times D_{17}$), $C_3\times C_{204}$ ($C_2\times C_{68}$), $C_3\times C_{204}$ ($C_4\times D_{17}$), $C_3\times C_{204}$ ($C_2^2\times C_{34}$), $C_3\times C_{204}$ ($C_2\times D_{34}$)

Order 578: $C_{17}\times C_{34}$ ($C_{12}\times D_6$)

Order 408: $C_{408}$ ($C_3\times D_{34}$) x 2, $C_{408}$ ($C_2\times C_{102}$) x 2, $C_{408}$ ($D_{102}$) x 2, $C_{408}$ ($S_3\times C_{34}$) x 2, $C_{51}:C_8$ ($C_2\times C_{102}$) x 2, $C_{51}:D_4$ ($C_{204}$) x 2, $C_4\times D_{51}$ ($C_2\times C_{102}$) x 2, $C_2\times C_{204}$ ($C_3\times D_{34}$), $C_2\times C_{204}$ ($C_2\times C_{102}$), $C_2\times C_{204}$ ($D_{102}$), $C_2\times C_{204}$ ($S_3\times C_{34}$), $D_{204}$ ($C_{204}$), $C_{51}:Q_8$ ($C_{204}$)

Order 306: $C_3\times C_{102}$ ($C_2^2\times C_{68}$), $C_3\times C_{102}$ ($C_4\times D_{34}$)

Order 289: $C_{17}^2$ ($C_{24}.D_6$)

Order 272: $C_2\times C_{136}$ ($C_3\times D_{51}$), $C_2\times C_{136}$ ($S_3\times C_{51}$)

Order 204: $C_{51}:C_4$ ($C_2\times C_{204}$) x 2, $D_{102}$ ($C_2\times C_{204}$) x 2, $C_{204}$ ($C_2^2\times C_{102}$), $C_{204}$ ($C_2\times D_{102}$), $C_{204}$ ($D_6\times C_{34}$), $C_{204}$ ($C_6\times D_{34}$), $C_{204}$ ($C_2\times C_{204}$), $C_{204}$ ($C_4\times D_{51}$), $C_{204}$ ($S_3\times C_{68}$), $C_{204}$ ($C_{12}\times D_{17}$), $C_2\times C_{102}$ ($C_2\times C_{204}$), $C_2\times C_{102}$ ($C_4\times D_{51}$), $C_2\times C_{102}$ ($S_3\times C_{68}$), $C_2\times C_{102}$ ($C_{12}\times D_{17}$)

Order 153: $C_3\times C_{51}$ ($\OD_{16}:C_{34}$), $C_3\times C_{51}$ ($C_8.D_{34}$)

Order 144: $C_6\times C_{24}$ ($C_{17}\times D_{17}$)

Order 136: $C_{136}$ ($C_3\times D_{102}$) x 2, $C_{136}$ ($S_3\times C_{102}$) x 2, $C_2\times C_{68}$ ($C_3\times D_{102}$), $C_2\times C_{68}$ ($S_3\times C_{102}$)

Order 102: $C_{102}$ ($C_2^2\times C_{204}$), $C_{102}$ ($C_4\times D_{102}$), $C_{102}$ ($D_6\times C_{68}$), $C_{102}$ ($C_{12}\times D_{34}$)

Order 72: $C_3\times C_{24}$ ($C_{17}\times D_{34}$) x 2, $C_6\times C_{12}$ ($C_{17}\times D_{34}$)

Order 68: $C_2\times C_{34}$ ($S_3\times C_{204}$), $C_2\times C_{34}$ ($C_{12}\times D_{51}$), $C_{68}$ ($S_3\times C_{204}$), $C_{68}$ ($C_6\times D_{102}$), $C_{68}$ ($D_6\times C_{102}$), $C_{68}$ ($C_{12}\times D_{51}$)

Order 51: $C_{51}$ ($\OD_{16}:C_{102}$), $C_{51}$ ($C_8.D_{102}$), $C_{51}$ ($D_{12}.C_{68}$), $C_{51}$ ($C_{24}.D_{34}$)

Order 48: $C_2\times C_{24}$ ($C_{17}\times D_{51}$), $C_2\times C_{24}$ ($D_{17}\times C_{51}$)

Order 36: $C_3\times C_{12}$ ($C_{34}\times D_{34}$), $C_3\times C_{12}$ ($D_{17}\times C_{68}$), $C_6^2$ ($D_{17}\times C_{68}$)

Order 34: $C_{34}$ ($C_{204}\times D_6$), $C_{34}$ ($C_{12}\times D_{102}$)

Order 24: $C_{24}$ ($C_{51}\times D_{34}$) x 2, $C_{24}$ ($C_{17}\times D_{102}$) x 2, $C_2\times C_{12}$ ($C_{51}\times D_{34}$), $C_2\times C_{12}$ ($C_{17}\times D_{102}$)

Order 18: $C_3\times C_6$ ($C_{68}\times D_{34}$)

Order 17: $C_{17}$ ($C_{24}.D_{102}$), $C_{17}$ ($D_{12}.C_{204}$)

Order 16: $C_2\times C_8$ ($C_{51}\times D_{51}$)

Order 12: $C_2\times C_6$ ($C_{68}\times D_{51}$), $C_2\times C_6$ ($D_{17}\times C_{204}$), $C_{12}$ ($C_{34}\times D_{102}$), $C_{12}$ ($C_{102}\times D_{34}$), $C_{12}$ ($C_{68}\times D_{51}$), $C_{12}$ ($D_{17}\times C_{204}$)

Order 9: $C_3^2$ ($C_{136}.D_{34}$)

Order 8: $C_8$ ($C_{102}\times D_{51}$) x 2, $C_2\times C_4$ ($C_{102}\times D_{51}$)

Order 6: $C_6$ ($C_{68}\times D_{102}$), $C_6$ ($C_{204}\times D_{34}$)

Order 4: $C_2^2$ ($D_{51}\times C_{204}$), $C_4$ ($C_{102}\times D_{102}$), $C_4$ ($D_{51}\times C_{204}$)

Order 3: $C_3$ ($D_{68}.C_{204}$), $C_3$ ($C_{136}.D_{102}$)

Order 2: $C_2$ ($C_{204}\times D_{102}$)

Order 1: $C_1$ ($C_{408}.D_{102}$)

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

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $21420 \times 21420$ character table is not available for this group.

Rational character table

The $431 \times 431$ rational character table is not available for this group.