Abstract group 226464.a: $D_{337}:C_{336}$

• Label: 226464.a
• Order: \( 2^{5} \cdot 3 \cdot 7 \cdot 337 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $353$
• Trans deg.: $5392$
• Rank: $2$

Group information

Description:	$D_{337}:C_{336}$
Order:	\(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \)
Exponent:	\(113232\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 337 \)
Automorphism group:	$C_{674}.C_{336}.C_2^2$, of order \(905856\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$, $C_{337}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336	337	674	1348
Elements	1	675	674	676	2022	2022	2696	2696	6066	5392	4044	5392	8088	12132	10784	16176	16176	32352	32352	64704	336	336	672	226464
Conjugacy classes	1	3	2	4	6	6	8	8	18	16	12	16	24	36	32	48	48	96	96	192	2	2	4	680
Divisions	1	3	1	2	3	1	2	2	3	2	1	2	2	3	2	2	2	2	2	2	1	1	1	43
Autjugacy classes	1	2	2	3	4	6	4	6	12	4	12	8	18	24	8	24	36	24	48	48	1	1	1	297

Dimension	1	2	4	6	8	12	16	24	48	96	168	336	672
Irr. complex chars.	672	0	0	0	0	0	0	0	0	0	8	0	0	680
Irr. rational chars.	4	6	4	4	4	6	2	4	4	2	0	2	1	43

Minimal presentations

Permutation degree:	$353$
Transitive degree:	$5392$
Rank:	$2$
Inequivalent generating pairs:	$18432$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	168	not computed	672
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{1348}=1, a^{168}=b^{674}, b^{a}=b^{113} \rangle$
Permutation group:	Degree $353$ $\langle(2,3)(4,9)(5,8)(6,10)(7,11)(12,32)(13,28)(14,31)(15,33)(16,29)(17,27)(18,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 221 & 0 \\ 0 & 30 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 226 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{337}$ $\,\rtimes\,$ $C_{336}$ (2)	$(D_{337}:C_{48})$ $\,\rtimes\,$ $C_7$	$(D_{337}:C_{112})$ $\,\rtimes\,$ $C_3$	$(D_{337}:C_{16})$ $\,\rtimes\,$ $C_{21}$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{674}$ . $C_{168}$	$C_{1348}$ . $C_{168}$	$(D_{337}:C_{56})$ . $C_6$	$(D_{337}:C_8)$ . $C_{42}$	all 34

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

Homology

Abelianization:	$C_{2} \times C_{336} \simeq C_{2} \times C_{16} \times C_{3} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 17920 subgroups in 112 conjugacy classes, 59 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{337}:C_{168}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_2\times C_{336}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{674}:C_{168}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1348}$	$G/\operatorname{Fit} \simeq$ $C_{168}$
Radical:	$R \simeq$ $D_{337}:C_{336}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{674}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{168}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
337-Sylow subgroup:	$P_{ 337 } \simeq$ $C_{337}$

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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Classes of subgroups up to conjugation

Order 226464: $D_{337}:C_{336}$

Order 113232: $C_{337}:C_{336}$ x 2, $D_{337}:C_{168}$

Order 75488: $D_{337}:C_{112}$

Order 56616: $C_{337}:C_{168}$ x 2, $C_{1348}:C_{42}$

Order 37744: $C_{337}:C_{112}$ x 2, $D_{337}:C_{56}$

Order 32352: $D_{337}:C_{48}$

Order 28308: $C_{1348}:C_{21}$, $C_{674}:C_{42}$, $C_{337}:C_{84}$

Order 18872: $C_{337}:C_{56}$ x 2, $C_{1348}:C_{14}$

Order 16176: $C_{337}:C_{48}$ x 2, $D_{337}:C_{24}$

Order 14154: $C_{337}:C_{42}$ x 2, $C_{337}:C_{42}$

Order 10784: $D_{337}:C_{16}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 8088: $C_{337}:C_{24}$ x 2, $C_{1348}:C_6$

Order 7077: $C_{337}:C_{21}$

Order 5392: $C_{337}:C_{16}$ x 2, $D_{337}:C_8$

Order 4718: $C_{337}:C_{14}$ x 2, $C_{337}:C_{14}$

Order 4044: $C_{337}:C_{12}$, $C_{674}:C_6$, $C_{1348}:C_3$

Order 2696: $C_{337}:C_8$ x 2, $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{337}:C_6$ x 2, $C_{337}:C_6$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 1011: $C_{337}:C_3$

Order 674: $D_{337}$ x 2, $C_{674}$

Order 672: $C_2\times C_{336}$

Order 337: $C_{337}$

Order 336: $C_{336}$ x 2, $C_2\times C_{168}$

Order 224: $C_2\times C_{112}$

Order 168: $C_{168}$ x 2, $C_2\times C_{84}$

Order 112: $C_{112}$ x 2, $C_2\times C_{56}$

Order 96: $C_2\times C_{48}$

Order 84: $C_{84}$ x 2, $C_2\times C_{42}$

Order 56: $C_{56}$ x 2, $C_2\times C_{28}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 42: $C_{42}$ x 3

Order 32: $C_2\times C_{16}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 21: $C_{21}$

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

Order 14: $C_{14}$ x 3

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

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 226464: $D_{337}:C_{336}$

Order 113232: $D_{337}:C_{168}$, $C_{337}:C_{336}$

Order 75488: $D_{337}:C_{112}$

Order 56616: $C_{337}:C_{168}$ x 2, $C_{1348}:C_{42}$

Order 37744: $D_{337}:C_{56}$, $C_{337}:C_{112}$

Order 32352: $D_{337}:C_{48}$

Order 28308: $C_{1348}:C_{21}$, $C_{674}:C_{42}$, $C_{337}:C_{84}$

Order 18872: $C_{337}:C_{56}$ x 2, $C_{1348}:C_{14}$

Order 16176: $C_{337}:C_{48}$, $D_{337}:C_{24}$

Order 14154: $C_{337}:C_{42}$, $C_{337}:C_{42}$

Order 10784: $D_{337}:C_{16}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 8088: $C_{337}:C_{24}$ x 2, $C_{1348}:C_6$

Order 7077: $C_{337}:C_{21}$

Order 5392: $C_{337}:C_{16}$, $D_{337}:C_8$

Order 4718: $C_{337}:C_{14}$, $C_{337}:C_{14}$

Order 4044: $C_{337}:C_{12}$, $C_{674}:C_6$, $C_{1348}:C_3$

Order 2696: $C_{337}:C_8$ x 2, $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 2022: $C_{337}:C_6$, $C_{337}:C_6$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 1011: $C_{337}:C_3$

Order 674: $C_{674}$, $D_{337}$

Order 672: $C_2\times C_{336}$

Order 337: $C_{337}$

Order 336: $C_{336}$, $C_2\times C_{168}$

Order 224: $C_2\times C_{112}$

Order 168: $C_{168}$ x 2, $C_2\times C_{84}$

Order 112: $C_2\times C_{56}$, $C_{112}$

Order 96: $C_2\times C_{48}$

Order 84: $C_{84}$ x 2, $C_2\times C_{42}$

Order 56: $C_{56}$ x 2, $C_2\times C_{28}$

Order 48: $C_2\times C_{24}$, $C_{48}$

Order 42: $C_{42}$ x 2

Order 32: $C_2\times C_{16}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 21: $C_{21}$

Order 16: $C_2\times C_8$, $C_{16}$

Order 14: $C_{14}$ x 2

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

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

Order 7: $C_7$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 226464: $D_{337}:C_{336}$ ($C_1$)

Order 113232: $C_{337}:C_{336}$ ($C_2$) x 2, $D_{337}:C_{168}$ ($C_2$)

Order 75488: $D_{337}:C_{112}$ ($C_3$)

Order 56616: $C_{1348}:C_{42}$ ($C_4$), $C_{337}:C_{168}$ ($C_2^2$), $C_{337}:C_{168}$ ($C_4$)

Order 37744: $C_{337}:C_{112}$ ($C_6$) x 2, $D_{337}:C_{56}$ ($C_6$)

Order 32352: $D_{337}:C_{48}$ ($C_7$)

Order 28308: $C_{1348}:C_{21}$ ($C_8$), $C_{674}:C_{42}$ ($C_8$), $C_{337}:C_{84}$ ($C_2\times C_4$)

Order 18872: $C_{1348}:C_{14}$ ($C_{12}$), $C_{337}:C_{56}$ ($C_2\times C_6$), $C_{337}:C_{56}$ ($C_{12}$)

Order 16176: $C_{337}:C_{48}$ ($C_{14}$) x 2, $D_{337}:C_{24}$ ($C_{14}$)

Order 14154: $C_{337}:C_{42}$ ($C_{16}$) x 2, $C_{337}:C_{42}$ ($C_2\times C_8$)

Order 10784: $D_{337}:C_{16}$ ($C_{21}$)

Order 9436: $C_{337}:C_{28}$ ($C_2\times C_{12}$), $C_{674}:C_{14}$ ($C_{24}$), $C_{1348}:C_7$ ($C_{24}$)

Order 8088: $C_{337}:C_{24}$ ($C_2\times C_{14}$), $C_{337}:C_{24}$ ($C_{28}$), $C_{1348}:C_6$ ($C_{28}$)

Order 7077: $C_{337}:C_{21}$ ($C_2\times C_{16}$)

Order 5392: $C_{337}:C_{16}$ ($C_{42}$) x 2, $D_{337}:C_8$ ($C_{42}$)

Order 4718: $C_{337}:C_{14}$ ($C_{48}$) x 2, $C_{337}:C_{14}$ ($C_2\times C_{24}$)

Order 4044: $C_{337}:C_{12}$ ($C_2\times C_{28}$), $C_{674}:C_6$ ($C_{56}$), $C_{1348}:C_3$ ($C_{56}$)

Order 2696: $C_4\times D_{337}$ ($C_{84}$), $C_{337}:C_8$ ($C_{84}$), $C_{337}:C_8$ ($C_2\times C_{42}$)

Order 2359: $C_{337}:C_7$ ($C_2\times C_{48}$)

Order 2022: $C_{337}:C_6$ ($C_{112}$) x 2, $C_{337}:C_6$ ($C_2\times C_{56}$)

Order 1348: $D_{674}$ ($C_{168}$), $C_{1348}$ ($C_{168}$), $C_{337}:C_4$ ($C_2\times C_{84}$)

Order 1011: $C_{337}:C_3$ ($C_2\times C_{112}$)

Order 674: $D_{337}$ ($C_{336}$) x 2, $C_{674}$ ($C_2\times C_{168}$)

Order 337: $C_{337}$ ($C_2\times C_{336}$)

Order 4: $C_4$ ($C_{337}:C_{168}$)

Order 2: $C_2$ ($C_{674}:C_{168}$)

Order 1: $C_1$ ($D_{337}:C_{336}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 226464: $D_{337}:C_{336}$ ($C_1$)

Order 113232: $D_{337}:C_{168}$ ($C_2$), $C_{337}:C_{336}$ ($C_2$)

Order 75488: $D_{337}:C_{112}$ ($C_3$)

Order 56616: $C_{1348}:C_{42}$ ($C_4$), $C_{337}:C_{168}$ ($C_2^2$), $C_{337}:C_{168}$ ($C_4$)

Order 37744: $D_{337}:C_{56}$ ($C_6$), $C_{337}:C_{112}$ ($C_6$)

Order 32352: $D_{337}:C_{48}$ ($C_7$)

Order 28308: $C_{1348}:C_{21}$ ($C_8$), $C_{674}:C_{42}$ ($C_8$), $C_{337}:C_{84}$ ($C_2\times C_4$)

Order 18872: $C_{1348}:C_{14}$ ($C_{12}$), $C_{337}:C_{56}$ ($C_2\times C_6$), $C_{337}:C_{56}$ ($C_{12}$)

Order 16176: $C_{337}:C_{48}$ ($C_{14}$), $D_{337}:C_{24}$ ($C_{14}$)

Order 14154: $C_{337}:C_{42}$ ($C_{16}$), $C_{337}:C_{42}$ ($C_2\times C_8$)

Order 10784: $D_{337}:C_{16}$ ($C_{21}$)

Order 9436: $C_{337}:C_{28}$ ($C_2\times C_{12}$), $C_{674}:C_{14}$ ($C_{24}$), $C_{1348}:C_7$ ($C_{24}$)

Order 8088: $C_{337}:C_{24}$ ($C_2\times C_{14}$), $C_{337}:C_{24}$ ($C_{28}$), $C_{1348}:C_6$ ($C_{28}$)

Order 7077: $C_{337}:C_{21}$ ($C_2\times C_{16}$)

Order 5392: $C_{337}:C_{16}$ ($C_{42}$), $D_{337}:C_8$ ($C_{42}$)

Order 4718: $C_{337}:C_{14}$ ($C_2\times C_{24}$), $C_{337}:C_{14}$ ($C_{48}$)

Order 4044: $C_{337}:C_{12}$ ($C_2\times C_{28}$), $C_{674}:C_6$ ($C_{56}$), $C_{1348}:C_3$ ($C_{56}$)

Order 2696: $C_4\times D_{337}$ ($C_{84}$), $C_{337}:C_8$ ($C_{84}$), $C_{337}:C_8$ ($C_2\times C_{42}$)

Order 2359: $C_{337}:C_7$ ($C_2\times C_{48}$)

Order 2022: $C_{337}:C_6$ ($C_2\times C_{56}$), $C_{337}:C_6$ ($C_{112}$)

Order 1348: $D_{674}$ ($C_{168}$), $C_{1348}$ ($C_{168}$), $C_{337}:C_4$ ($C_2\times C_{84}$)

Order 1011: $C_{337}:C_3$ ($C_2\times C_{112}$)

Order 674: $C_{674}$ ($C_2\times C_{168}$), $D_{337}$ ($C_{336}$)

Order 337: $C_{337}$ ($C_2\times C_{336}$)

Order 4: $C_4$ ($C_{337}:C_{168}$)

Order 2: $C_2$ ($C_{674}:C_{168}$)

Order 1: $C_1$ ($D_{337}:C_{336}$)

Series

Derived series

$D_{337}:C_{336}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$D_{337}:C_{336}$

$\rhd$

$D_{337}:C_{168}$

$\rhd$

$C_{1348}:C_{42}$

$\rhd$

$C_{1348}:C_{21}$

$\rhd$

$C_{1348}:C_7$

$\rhd$

$C_{1348}$

$\rhd$

$C_{674}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$D_{337}:C_{336}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

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

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

Character theory

Complex character table

The $680 \times 680$ character table is not available for this group.

Rational character table

See the $43 \times 43$ rational character table.