Abstract group 4896.q: $D_{12}.C_{204}$

• Label: 4896.q
• Order: \( 2^{5} \cdot 3^{2} \cdot 17 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \cdot 17 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $39$
• Trans deg.: $816$
• Rank: $3$

Group information

Description:	$D_{12}.C_{204}$
Order:	\(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \)
Exponent:	\(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \)
Automorphism group:	$(C_5\times D_5^2):F_5$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_{17}$
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	15	8	16	48	32	56	16	112	240	128	256	768	512	896	1792	4896
Conjugacy classes	1	4	5	5	17	10	22	16	44	64	80	80	272	160	352	704	1836
Divisions	1	4	3	4	9	4	8	1	8	4	3	4	9	4	8	8	82
Autjugacy classes	1	3	3	3	7	3	7	1	7	3	3	3	7	3	7	7	68

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$816$
Rank:	$3$
Inequivalent generating triples:	$5363904$

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

Presentation:	$\langle a, b, c \mid a^{2}=b^{408}=c^{6}=[a,b]=[b,c]=1, c^{a}=b^{204}c^{5} \rangle$
Permutation group:	Degree $39$ $\langle(1,2)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13)(15,16)(18,19), (1,3,6,2)(4,7,10,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 360 & 0 \\ 0 & 217 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{409})$
Direct product:	$C_3$ $\, \times\, $ $C_{17}$ $\, \times\, $ $(C_8.D_6)$
Semidirect product:	$(S_3\times C_{408})$ $\,\rtimes\,$ $C_2$ (2)	$(C_2\times C_{408})$ $\,\rtimes\,$ $S_3$	$(S_3\times C_{136})$ $\,\rtimes\,$ $C_6$ (2)	$(S_3\times C_{24})$ $\,\rtimes\,$ $C_{34}$ (2)	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{408}$ . $D_6$ (2)	$D_{12}$ . $C_{204}$	$C_{68}$ . $(C_6\times D_6)$	$C_6$ . $(D_6\times C_{68})$	all 80

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:	$1$

Subgroups

There are 500 subgroups in 270 conjugacy classes, 148 normal (92 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{408}$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^2\times C_{204}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times C_{102}$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{408}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{12}.C_{204}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{102}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4896: $D_{12}.C_{204}$

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

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

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

Order 816: $C_2\times C_{408}$ x 5, $\OD_{16}\times C_{51}$ x 3, $C_{24}:C_{34}$ x 2, $S_3\times C_{136}$ x 2, $C_{12}.C_{68}$, $D_4:C_{102}$, $D_{12}:C_{34}$

Order 612: $S_3\times C_{102}$ x 2, $C_3\times C_{204}$ x 2, $C_3:C_{204}$ x 2, $C_6\times C_{102}$

Order 544: $\OD_{16}:C_{34}$

Order 408: $C_{408}$ x 8, $C_2\times C_{204}$ x 5, $D_4\times C_{51}$ x 3, $D_6:C_{34}$ x 2, $S_3\times C_{68}$ x 2, $C_3:C_{136}$ x 2, $Q_8\times C_{51}$, $C_{17}\times D_{12}$, $C_{51}:Q_8$

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

Order 288: $C_{24}.D_6$

Order 272: $\OD_{16}\times C_{17}$ x 3, $C_2\times C_{136}$ x 3, $D_4:C_{34}$

Order 204: $C_{204}$ x 8, $C_2\times C_{102}$ x 5, $S_3\times C_{34}$ x 2, $C_3:C_{68}$ x 2

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

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 4, $C_2\times C_{68}$ x 3, $D_4\times C_{17}$ x 3, $Q_8\times C_{17}$

Order 102: $C_{102}$ x 9, $S_3\times C_{17}$ 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 4, $C_2\times C_{34}$ x 3

Order 51: $C_{51}$ x 3

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 4

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: $C_{17}$

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

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 4, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 9, $S_3$ x 2

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4896: $D_{12}.C_{204}$

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

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

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

Order 816: $C_2\times C_{408}$ x 4, $\OD_{16}\times C_{51}$ x 2, $C_{12}.C_{68}$, $C_{24}:C_{34}$, $S_3\times C_{136}$, $D_4:C_{102}$, $D_{12}:C_{34}$

Order 612: $C_3\times C_{204}$ x 2, $C_6\times C_{102}$, $S_3\times C_{102}$, $C_3:C_{204}$

Order 544: $\OD_{16}:C_{34}$

Order 408: $C_{408}$ x 7, $C_2\times C_{204}$ x 4, $D_4\times C_{51}$ x 2, $Q_8\times C_{51}$, $D_6:C_{34}$, $C_{17}\times D_{12}$, $S_3\times C_{68}$, $C_{51}:Q_8$, $C_3:C_{136}$

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

Order 288: $C_{24}.D_6$

Order 272: $\OD_{16}\times C_{17}$ x 2, $C_2\times C_{136}$ x 2, $D_4:C_{34}$

Order 204: $C_{204}$ x 7, $C_2\times C_{102}$ x 4, $S_3\times C_{34}$, $C_3:C_{68}$

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

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

Order 136: $C_{136}$ x 3, $C_2\times C_{68}$ x 2, $D_4\times C_{17}$ x 2, $Q_8\times C_{17}$

Order 102: $C_{102}$ x 7, $S_3\times C_{17}$

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

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

Order 68: $C_{68}$ x 3, $C_2\times C_{34}$ x 2

Order 51: $C_{51}$ x 3

Order 48: $C_2\times C_{24}$ x 4, $C_3\times \OD_{16}$ x 2, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3:\OD_{16}$

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

Order 34: $C_{34}$ x 3

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

Order 24: $C_{24}$ x 7, $C_2\times C_{12}$ x 4, $C_3\times D_4$ 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: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 17: $C_{17}$

Order 16: $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $D_4:C_2$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 7, $S_3$

Order 4: $C_4$ x 3, $C_2^2$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4896: $D_{12}.C_{204}$ ($C_1$)

Order 2448: $C_{24}:C_{102}$ ($C_2$) x 2, $S_3\times C_{408}$ ($C_2$) x 2, $C_6\times C_{408}$ ($C_2$), $C_{12}.C_{204}$ ($C_2$), $D_{12}:C_{102}$ ($C_2$)

Order 1632: $D_{12}.C_{68}$ ($C_3$)

Order 1224: $S_3\times C_{204}$ ($C_2^2$) x 2, $C_3\times C_{408}$ ($C_2^2$) x 2, $C_3:C_{408}$ ($C_2^2$) x 2, $D_6:C_{102}$ ($C_4$) x 2, $D_{12}\times C_{51}$ ($C_4$), $C_{12}.C_{102}$ ($C_4$), $C_6\times C_{204}$ ($C_2^2$)

Order 816: $C_{24}:C_{34}$ ($C_6$) x 2, $S_3\times C_{136}$ ($C_6$) x 2, $C_2\times C_{408}$ ($C_6$), $C_2\times C_{408}$ ($S_3$), $C_{12}.C_{68}$ ($C_6$), $D_{12}:C_{34}$ ($C_6$)

Order 612: $S_3\times C_{102}$ ($C_2\times C_4$) x 2, $C_3:C_{204}$ ($C_2\times C_4$) x 2, $C_6\times C_{102}$ ($C_2\times C_4$), $C_3\times C_{204}$ ($C_2^3$), $C_3\times C_{204}$ ($C_2\times C_4$)

Order 408: $C_{408}$ ($C_2\times C_6$) x 2, $C_{408}$ ($D_6$) x 2, $D_6:C_{34}$ ($C_{12}$) x 2, $S_3\times C_{68}$ ($C_2\times C_6$) x 2, $C_3:C_{136}$ ($C_2\times C_6$) x 2, $C_2\times C_{204}$ ($C_2\times C_6$), $C_2\times C_{204}$ ($D_6$), $C_{17}\times D_{12}$ ($C_{12}$), $C_{51}:Q_8$ ($C_{12}$)

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

Order 288: $C_{24}.D_6$ ($C_{17}$)

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

Order 204: $S_3\times C_{34}$ ($C_2\times C_{12}$) x 2, $C_3:C_{68}$ ($C_2\times C_{12}$) x 2, $C_{204}$ ($C_2\times C_{12}$), $C_{204}$ ($C_4\times S_3$), $C_{204}$ ($C_2^2\times C_6$), $C_{204}$ ($C_2\times D_6$), $C_2\times C_{102}$ ($C_2\times C_{12}$), $C_2\times C_{102}$ ($C_4\times S_3$)

Order 153: $C_3\times C_{51}$ ($\OD_{16}:C_2$)

Order 144: $C_{24}:C_6$ ($C_{34}$) x 2, $S_3\times C_{24}$ ($C_{34}$) x 2, $C_{12}.C_{12}$ ($C_{34}$), $D_{12}:C_6$ ($C_{34}$), $C_6\times C_{24}$ ($C_{34}$)

Order 136: $C_{136}$ ($C_6\times S_3$) x 2, $C_2\times C_{68}$ ($C_6\times S_3$)

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

Order 96: $C_8.D_6$ ($C_{51}$)

Order 72: $C_6\wr C_2$ ($C_{68}$) x 2, $S_3\times C_{12}$ ($C_2\times C_{34}$) x 2, $C_3\times C_{24}$ ($C_2\times C_{34}$) x 2, $C_3:C_{24}$ ($C_2\times C_{34}$) x 2, $C_6\times C_{12}$ ($C_2\times C_{34}$), $C_3\times D_{12}$ ($C_{68}$), $C_3^2:Q_8$ ($C_{68}$)

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

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

Order 48: $C_{24}:C_2$ ($C_{102}$) x 2, $S_3\times C_8$ ($C_{102}$) x 2, $D_{12}:C_2$ ($C_{102}$), $C_2\times C_{24}$ ($C_{102}$), $C_2\times C_{24}$ ($S_3\times C_{17}$), $C_3:\OD_{16}$ ($C_{102}$)

Order 36: $C_3:C_{12}$ ($C_2\times C_{68}$) x 2, $C_6\times S_3$ ($C_2\times C_{68}$) x 2, $C_3\times C_{12}$ ($C_2\times C_{68}$), $C_3\times C_{12}$ ($C_2^2\times C_{34}$), $C_6^2$ ($C_2\times C_{68}$)

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

Order 24: $C_3:D_4$ ($C_{204}$) x 2, $C_4\times S_3$ ($C_2\times C_{102}$) x 2, $C_{24}$ ($C_2\times C_{102}$) x 2, $C_{24}$ ($S_3\times C_{34}$) x 2, $C_3:C_8$ ($C_2\times C_{102}$) x 2, $C_2\times C_{12}$ ($C_2\times C_{102}$), $C_2\times C_{12}$ ($S_3\times C_{34}$), $D_{12}$ ($C_{204}$), $C_3:Q_8$ ($C_{204}$)

Order 18: $C_3\times C_6$ ($C_2^2\times C_{68}$)

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

Order 16: $C_2\times C_8$ ($S_3\times C_{51}$)

Order 12: $D_6$ ($C_2\times C_{204}$) x 2, $C_3:C_4$ ($C_2\times C_{204}$) x 2, $C_2\times C_6$ ($C_2\times C_{204}$), $C_2\times C_6$ ($S_3\times C_{68}$), $C_{12}$ ($C_2^2\times C_{102}$), $C_{12}$ ($D_6\times C_{34}$), $C_{12}$ ($C_2\times C_{204}$), $C_{12}$ ($S_3\times C_{68}$)

Order 9: $C_3^2$ ($\OD_{16}:C_{34}$)

Order 8: $C_8$ ($S_3\times C_{102}$) x 2, $C_2\times C_4$ ($S_3\times C_{102}$)

Order 6: $C_6$ ($C_2^2\times C_{204}$), $C_6$ ($D_6\times C_{68}$)

Order 4: $C_2^2$ ($S_3\times C_{204}$), $C_4$ ($S_3\times C_{204}$), $C_4$ ($D_6\times C_{102}$)

Order 3: $C_3$ ($\OD_{16}:C_{102}$), $C_3$ ($D_{12}.C_{68}$)

Order 2: $C_2$ ($C_{204}\times D_6$)

Order 1: $C_1$ ($D_{12}.C_{204}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4896: $D_{12}.C_{204}$ ($C_1$)

Order 2448: $C_6\times C_{408}$ ($C_2$), $C_{24}:C_{102}$ ($C_2$), $S_3\times C_{408}$ ($C_2$), $C_{12}.C_{204}$ ($C_2$), $D_{12}:C_{102}$ ($C_2$)

Order 1632: $D_{12}.C_{68}$ ($C_3$)

Order 1224: $C_3\times C_{408}$ ($C_2^2$) x 2, $D_{12}\times C_{51}$ ($C_4$), $S_3\times C_{204}$ ($C_2^2$), $C_{12}.C_{102}$ ($C_4$), $C_3:C_{408}$ ($C_2^2$), $C_6\times C_{204}$ ($C_2^2$), $D_6:C_{102}$ ($C_4$)

Order 816: $C_2\times C_{408}$ ($C_6$), $C_2\times C_{408}$ ($S_3$), $C_{12}.C_{68}$ ($C_6$), $C_{24}:C_{34}$ ($C_6$), $S_3\times C_{136}$ ($C_6$), $D_{12}:C_{34}$ ($C_6$)

Order 612: $C_6\times C_{102}$ ($C_2\times C_4$), $S_3\times C_{102}$ ($C_2\times C_4$), $C_3\times C_{204}$ ($C_2^3$), $C_3\times C_{204}$ ($C_2\times C_4$), $C_3:C_{204}$ ($C_2\times C_4$)

Order 408: $C_{408}$ ($C_2\times C_6$) x 2, $C_{408}$ ($D_6$) x 2, $C_2\times C_{204}$ ($C_2\times C_6$), $C_2\times C_{204}$ ($D_6$), $D_6:C_{34}$ ($C_{12}$), $C_{17}\times D_{12}$ ($C_{12}$), $S_3\times C_{68}$ ($C_2\times C_6$), $C_{51}:Q_8$ ($C_{12}$), $C_3:C_{136}$ ($C_2\times C_6$)

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

Order 288: $C_{24}.D_6$ ($C_{17}$)

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

Order 204: $C_{204}$ ($C_2\times C_{12}$), $C_{204}$ ($C_4\times S_3$), $C_{204}$ ($C_2^2\times C_6$), $C_{204}$ ($C_2\times D_6$), $C_2\times C_{102}$ ($C_2\times C_{12}$), $C_2\times C_{102}$ ($C_4\times S_3$), $S_3\times C_{34}$ ($C_2\times C_{12}$), $C_3:C_{68}$ ($C_2\times C_{12}$)

Order 153: $C_3\times C_{51}$ ($\OD_{16}:C_2$)

Order 144: $C_{12}.C_{12}$ ($C_{34}$), $C_{24}:C_6$ ($C_{34}$), $S_3\times C_{24}$ ($C_{34}$), $D_{12}:C_6$ ($C_{34}$), $C_6\times C_{24}$ ($C_{34}$)

Order 136: $C_{136}$ ($C_6\times S_3$) x 2, $C_2\times C_{68}$ ($C_6\times S_3$)

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

Order 96: $C_8.D_6$ ($C_{51}$)

Order 72: $C_3\times C_{24}$ ($C_2\times C_{34}$) x 2, $C_6\times C_{12}$ ($C_2\times C_{34}$), $C_6\wr C_2$ ($C_{68}$), $C_3\times D_{12}$ ($C_{68}$), $S_3\times C_{12}$ ($C_2\times C_{34}$), $C_3^2:Q_8$ ($C_{68}$), $C_3:C_{24}$ ($C_2\times C_{34}$)

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

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

Order 48: $C_{24}:C_2$ ($C_{102}$), $S_3\times C_8$ ($C_{102}$), $D_{12}:C_2$ ($C_{102}$), $C_2\times C_{24}$ ($C_{102}$), $C_2\times C_{24}$ ($S_3\times C_{17}$), $C_3:\OD_{16}$ ($C_{102}$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_{68}$), $C_3\times C_{12}$ ($C_2^2\times C_{34}$), $C_3:C_{12}$ ($C_2\times C_{68}$), $C_6^2$ ($C_2\times C_{68}$), $C_6\times S_3$ ($C_2\times C_{68}$)

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

Order 24: $C_{24}$ ($C_2\times C_{102}$) x 2, $C_{24}$ ($S_3\times C_{34}$) x 2, $C_2\times C_{12}$ ($C_2\times C_{102}$), $C_2\times C_{12}$ ($S_3\times C_{34}$), $C_3:D_4$ ($C_{204}$), $D_{12}$ ($C_{204}$), $C_4\times S_3$ ($C_2\times C_{102}$), $C_3:Q_8$ ($C_{204}$), $C_3:C_8$ ($C_2\times C_{102}$)

Order 18: $C_3\times C_6$ ($C_2^2\times C_{68}$)

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

Order 16: $C_2\times C_8$ ($S_3\times C_{51}$)

Order 12: $C_2\times C_6$ ($C_2\times C_{204}$), $C_2\times C_6$ ($S_3\times C_{68}$), $D_6$ ($C_2\times C_{204}$), $C_{12}$ ($C_2^2\times C_{102}$), $C_{12}$ ($D_6\times C_{34}$), $C_{12}$ ($C_2\times C_{204}$), $C_{12}$ ($S_3\times C_{68}$), $C_3:C_4$ ($C_2\times C_{204}$)

Order 9: $C_3^2$ ($\OD_{16}:C_{34}$)

Order 8: $C_8$ ($S_3\times C_{102}$) x 2, $C_2\times C_4$ ($S_3\times C_{102}$)

Order 6: $C_6$ ($C_2^2\times C_{204}$), $C_6$ ($D_6\times C_{68}$)

Order 4: $C_2^2$ ($S_3\times C_{204}$), $C_4$ ($S_3\times C_{204}$), $C_4$ ($D_6\times C_{102}$)

Order 3: $C_3$ ($\OD_{16}:C_{102}$), $C_3$ ($D_{12}.C_{68}$)

Order 2: $C_2$ ($C_{204}\times D_6$)

Order 1: $C_1$ ($D_{12}.C_{204}$)

Series

Derived series

$D_{12}.C_{204}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$D_{12}.C_{204}$

$\rhd$

$C_6\times C_{408}$

$\rhd$

$C_6\times C_{204}$

$\rhd$

$C_3\times C_{204}$

$\rhd$

$C_3\times C_{102}$

$\rhd$

$C_{102}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{12}.C_{204}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{408}$

$\lhd$

$C_2\times C_{408}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $1836 \times 1836$ character table is not available for this group.

Rational character table

The $82 \times 82$ rational character table is not available for this group.