Abstract group 336.65: $C_{28}:C_{12}$

• Label: 336.65
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$C_{28}:C_{12}$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^3:D_4\times F_7$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	3	2	60	6	6	120	18	12	24	36	48	336
Conjugacy classes	1	3	2	12	6	3	24	9	6	12	18	24	120
Divisions	1	3	1	6	3	1	6	3	1	2	3	2	32
Autjugacy classes	1	2	1	2	2	1	2	2	1	1	2	1	18

Dimension	1	2	4	6	12	24
Irr. complex chars.	48	72	0	0	0	0	120
Irr. rational chars.	4	10	6	4	6	2	32

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	12

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{84}=1, b^{a}=b^{13} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(8,9,10,11)(12,13,15,14), (12,14,15,13), (16,17,18), (8,10)(9,11)(12,15)(13,14), (12,15)(13,14), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 4 & 1 & 6 \\ 10 & 6 & 4 \\ 6 & 9 & 3 \end{array}\right), \left(\begin{array}{rrr} 1 & 10 & 8 \\ 12 & 12 & 8 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}\right), \left(\begin{array}{rrr} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right), \left(\begin{array}{rrr} 6 & 0 & 0 \\ 12 & 7 & 1 \\ 11 & 2 & 8 \end{array}\right), \left(\begin{array}{rrr} 10 & 0 & 0 \\ 11 & 12 & 2 \\ 9 & 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{13})$
Direct product:	$C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_7:C_4)$
Semidirect product:	$C_{84}$ $\,\rtimes\,$ $C_4$ (2)	$C_{28}$ $\,\rtimes\,$ $C_{12}$ (2)	$C_{21}$ $\,\rtimes\,$ $C_4^2$	$C_7$ $\,\rtimes\,$ $(C_4\times C_{12})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_4\times D_7)$ (2)	$(C_2\times C_{12})$ . $D_7$	$(C_2\times C_6)$ . $D_{14}$	$C_2$ . $(C_{12}\times D_7)$ (2)	all 16

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

Homology

Abelianization:	$C_{4} \times C_{12} \simeq C_{4}^{2} \times C_{3}$
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 144 subgroups in 60 conjugacy classes, 46 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{12}$	$G/Z \simeq$ $D_7$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{28}:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 336: $C_{28}:C_{12}$

Order 168: $C_{14}:C_{12}$ x 2, $C_2\times C_{84}$

Order 112: $C_{28}:C_4$

Order 84: $C_7:C_{12}$ x 4, $C_{84}$ x 2, $C_2\times C_{42}$

Order 56: $C_{14}:C_4$ x 2, $C_2\times C_{28}$

Order 48: $C_4\times C_{12}$

Order 42: $C_{42}$ x 3

Order 28: $C_7:C_4$ x 4, $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 3

Order 21: $C_{21}$

Order 16: $C_4^2$

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 6, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 336: $C_{28}:C_{12}$

Order 168: $C_2\times C_{84}$, $C_{14}:C_{12}$

Order 112: $C_{28}:C_4$

Order 84: $C_{84}$, $C_7:C_{12}$, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$, $C_{14}:C_4$

Order 48: $C_4\times C_{12}$

Order 42: $C_{42}$ x 2

Order 28: $C_2\times C_{14}$, $C_{28}$, $C_7:C_4$

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

Order 21: $C_{21}$

Order 16: $C_4^2$

Order 14: $C_{14}$ x 2

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

Order 8: $C_2\times C_4$ x 2

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: $C_{28}:C_{12}$ ($C_1$)

Order 168: $C_{14}:C_{12}$ ($C_2$) x 2, $C_2\times C_{84}$ ($C_2$)

Order 112: $C_{28}:C_4$ ($C_3$)

Order 84: $C_7:C_{12}$ ($C_4$) x 4, $C_{84}$ ($C_4$) x 2, $C_2\times C_{42}$ ($C_2^2$)

Order 56: $C_{14}:C_4$ ($C_6$) x 2, $C_2\times C_{28}$ ($C_6$)

Order 42: $C_{42}$ ($C_2\times C_4$) x 3

Order 28: $C_7:C_4$ ($C_{12}$) x 4, $C_{28}$ ($C_{12}$) x 2, $C_2\times C_{14}$ ($C_2\times C_6$)

Order 24: $C_2\times C_{12}$ ($D_7$)

Order 21: $C_{21}$ ($C_4^2$)

Order 14: $C_{14}$ ($C_2\times C_{12}$) x 3

Order 12: $C_{12}$ ($C_7:C_4$) x 2, $C_2\times C_6$ ($D_{14}$)

Order 8: $C_2\times C_4$ ($C_3\times D_7$)

Order 7: $C_7$ ($C_4\times C_{12}$)

Order 6: $C_6$ ($C_4\times D_7$) x 2, $C_6$ ($C_{14}:C_4$)

Order 4: $C_4$ ($C_7:C_{12}$) x 2, $C_2^2$ ($C_3\times D_{14}$)

Order 3: $C_3$ ($C_{28}:C_4$)

Order 2: $C_2$ ($C_{12}\times D_7$) x 2, $C_2$ ($C_{14}:C_{12}$)

Order 1: $C_1$ ($C_{28}:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $C_{28}:C_{12}$ ($C_1$)

Order 168: $C_2\times C_{84}$ ($C_2$), $C_{14}:C_{12}$ ($C_2$)

Order 112: $C_{28}:C_4$ ($C_3$)

Order 84: $C_{84}$ ($C_4$), $C_7:C_{12}$ ($C_4$), $C_2\times C_{42}$ ($C_2^2$)

Order 56: $C_2\times C_{28}$ ($C_6$), $C_{14}:C_4$ ($C_6$)

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

Order 28: $C_2\times C_{14}$ ($C_2\times C_6$), $C_{28}$ ($C_{12}$), $C_7:C_4$ ($C_{12}$)

Order 24: $C_2\times C_{12}$ ($D_7$)

Order 21: $C_{21}$ ($C_4^2$)

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

Order 12: $C_2\times C_6$ ($D_{14}$), $C_{12}$ ($C_7:C_4$)

Order 8: $C_2\times C_4$ ($C_3\times D_7$)

Order 7: $C_7$ ($C_4\times C_{12}$)

Order 6: $C_6$ ($C_{14}:C_4$), $C_6$ ($C_4\times D_7$)

Order 4: $C_2^2$ ($C_3\times D_{14}$), $C_4$ ($C_7:C_{12}$)

Order 3: $C_3$ ($C_{28}:C_4$)

Order 2: $C_2$ ($C_{14}:C_{12}$), $C_2$ ($C_{12}\times D_7$)

Order 1: $C_1$ ($C_{28}:C_{12}$)

Series

Derived series

$C_{28}:C_{12}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$C_{28}:C_{12}$

$\rhd$

$C_{14}:C_{12}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{28}:C_{12}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{12}$

Supergroups

This group is a maximal subgroup of 71 larger groups in the database.

This group is a maximal quotient of 16 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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