Abstract group 336.109: $C_2\times C_{168}$

• Label: 336.109
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$C_2\times C_{168}$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{12}:C_2^4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	28	42	56	84	168
Elements	1	3	2	4	6	6	8	8	18	12	16	24	36	48	48	96	336
Conjugacy classes	1	3	2	4	6	6	8	8	18	12	16	24	36	48	48	96	336
Divisions	1	3	1	2	3	1	2	2	3	1	2	2	3	2	2	2	32
Autjugacy classes	1	2	1	2	2	1	1	2	2	1	1	2	2	1	2	1	24

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	336	0	0	0	0	0	0	0	336
Irr. rational chars.	4	6	4	4	2	6	4	2	32

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	12

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{168}=1 \rangle$
Permutation group:	Degree $20$ $\langle(3,10,6,8,4,9,5,7), (1,2), (11,13,12), (14,20,19,18,17,16,15), (3,6,4,5)(7,10,8,9), (3,4)(5,6)(7,8)(9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 214 & 0 \\ 0 & 273 \end{array}\right), \left(\begin{array}{rr} 129 & 0 \\ 0 & 209 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_2$ $\, \times\, $ $C_8$ $\, \times\, $ $C_3$ $\, \times\, $ $C_7$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{84}$ . $C_4$	$C_4$ . $C_{84}$	$C_{28}$ . $C_{12}$	$C_{12}$ . $C_{28}$	all 20

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

Homology

Primary decomposition:	$C_{2} \times C_{8} \times C_{3} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 44 subgroups, all normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{168}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{168}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times C_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{168}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{168}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

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

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

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

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

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

Order 42: $C_{42}$ x 3

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: $C_2\times C_8$

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 336: $C_2\times C_{168}$

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

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

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

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

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

Order 42: $C_{42}$ x 2

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

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

Order 21: $C_{21}$

Order 16: $C_2\times C_8$

Order 14: $C_{14}$ x 2

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

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

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 336: $C_2\times C_{168}$ ($C_1$)

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

Order 112: $C_2\times C_{56}$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$), $C_{84}$ ($C_4$), $C_2\times C_{42}$ ($C_4$)

Order 56: $C_{56}$ ($C_6$) x 2, $C_2\times C_{28}$ ($C_6$)

Order 48: $C_2\times C_{24}$ ($C_7$)

Order 42: $C_{42}$ ($C_8$) x 2, $C_{42}$ ($C_2\times C_4$)

Order 28: $C_2\times C_{14}$ ($C_{12}$), $C_{28}$ ($C_2\times C_6$), $C_{28}$ ($C_{12}$)

Order 24: $C_{24}$ ($C_{14}$) x 2, $C_2\times C_{12}$ ($C_{14}$)

Order 21: $C_{21}$ ($C_2\times C_8$)

Order 16: $C_2\times C_8$ ($C_{21}$)

Order 14: $C_{14}$ ($C_{24}$) x 2, $C_{14}$ ($C_2\times C_{12}$)

Order 12: $C_2\times C_6$ ($C_{28}$), $C_{12}$ ($C_2\times C_{14}$), $C_{12}$ ($C_{28}$)

Order 8: $C_8$ ($C_{42}$) x 2, $C_2\times C_4$ ($C_{42}$)

Order 7: $C_7$ ($C_2\times C_{24}$)

Order 6: $C_6$ ($C_{56}$) x 2, $C_6$ ($C_2\times C_{28}$)

Order 4: $C_2^2$ ($C_{84}$), $C_4$ ($C_{84}$), $C_4$ ($C_2\times C_{42}$)

Order 3: $C_3$ ($C_2\times C_{56}$)

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

Order 1: $C_1$ ($C_2\times C_{168}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $C_2\times C_{168}$ ($C_1$)

Order 168: $C_{168}$ ($C_2$), $C_2\times C_{84}$ ($C_2$)

Order 112: $C_2\times C_{56}$ ($C_3$)

Order 84: $C_{84}$ ($C_2^2$), $C_{84}$ ($C_4$), $C_2\times C_{42}$ ($C_4$)

Order 56: $C_2\times C_{28}$ ($C_6$), $C_{56}$ ($C_6$)

Order 48: $C_2\times C_{24}$ ($C_7$)

Order 42: $C_{42}$ ($C_2\times C_4$), $C_{42}$ ($C_8$)

Order 28: $C_2\times C_{14}$ ($C_{12}$), $C_{28}$ ($C_2\times C_6$), $C_{28}$ ($C_{12}$)

Order 24: $C_2\times C_{12}$ ($C_{14}$), $C_{24}$ ($C_{14}$)

Order 21: $C_{21}$ ($C_2\times C_8$)

Order 16: $C_2\times C_8$ ($C_{21}$)

Order 14: $C_{14}$ ($C_2\times C_{12}$), $C_{14}$ ($C_{24}$)

Order 12: $C_2\times C_6$ ($C_{28}$), $C_{12}$ ($C_2\times C_{14}$), $C_{12}$ ($C_{28}$)

Order 8: $C_2\times C_4$ ($C_{42}$), $C_8$ ($C_{42}$)

Order 7: $C_7$ ($C_2\times C_{24}$)

Order 6: $C_6$ ($C_2\times C_{28}$), $C_6$ ($C_{56}$)

Order 4: $C_2^2$ ($C_{84}$), $C_4$ ($C_{84}$), $C_4$ ($C_2\times C_{42}$)

Order 3: $C_3$ ($C_2\times C_{56}$)

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

Order 1: $C_1$ ($C_2\times C_{168}$)

Series

Derived series

$C_2\times C_{168}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{168}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{168}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{168}$

Supergroups

This group is a maximal subgroup of 114 larger groups in the database.

This group is a maximal quotient of 43 larger groups in the database.

Character theory

Complex character table

See the $336 \times 336$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.