Abstract group 336.150: $C_{12}:D_{14}$

• Label: 336.150
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $84$
• Rank: $3$

Group information

Description:	$C_{12}:D_{14}$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^4\times C_6)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	83	2	44	58	6	4	78	12	12	12	24	336
Conjugacy classes	1	7	1	2	3	3	1	9	3	3	3	6	42
Divisions	1	7	1	2	3	1	1	3	1	1	1	1	23
Autjugacy classes	1	4	1	2	2	1	1	2	1	1	1	1	18

Dimension	1	2	4	6	12	24
Irr. complex chars.	8	18	16	0	0	0	42
Irr. rational chars.	8	6	1	4	3	1	23

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$84$
Rank:	$3$
Inequivalent generating triples:	$2688$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	24
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{84}=[a,b]=1, c^{a}=c^{71}, c^{b}=c^{55} \rangle$
Permutation group:	Degree $14$ $\langle(2,4)(9,10)(11,12)(13,14), (2,4)(6,7), (1,2,3,4), (1,3)(2,4), (5,6,7), (8,9,11,13,14,12,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 14 \\ 14 & 29 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 21 \\ 14 & 41 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 21 & 22 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{28}$ $\,\rtimes\,$ $S_3$	$D_{21}$ $\,\rtimes\,$ $D_4$ (2)	$D_{14}$ $\,\rtimes\,$ $D_6$ (2)	$D_{12}$ $\,\rtimes\,$ $D_7$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{42}$ . $C_2^2$	$C_{42}$ . $C_2^3$	$C_2$ . $(S_3\times D_{14})$	$C_{14}$ . $(C_2\times D_6)$	all 5

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 756 subgroups in 108 conjugacy classes, 34 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times D_{14}$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}:D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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

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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 336: $C_{12}:D_{14}$

Order 168: $S_3\times D_{14}$ x 2, $C_{21}:D_4$ x 2, $C_4\times D_{21}$, $C_7\times D_{12}$, $C_3\times D_{28}$

Order 112: $D_4\times D_7$

Order 84: $S_3\times D_7$ x 4, $S_3\times C_{14}$ x 2, $C_3\times D_{14}$ x 2, $C_{84}$, $C_{21}:C_4$, $D_{42}$

Order 56: $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $D_{28}$, $C_4\times D_7$

Order 48: $S_3\times D_4$

Order 42: $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 28: $D_{14}$ x 7, $C_2\times C_{14}$ x 2, $C_{28}$, $C_7:C_4$

Order 24: $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 21: $C_{21}$

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $D_6$ x 7, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 336: $C_{12}:D_{14}$

Order 168: $S_3\times D_{14}$, $C_4\times D_{21}$, $C_7\times D_{12}$, $C_3\times D_{28}$, $C_{21}:D_4$

Order 112: $D_4\times D_7$

Order 84: $S_3\times D_7$, $C_{84}$, $C_{21}:C_4$, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 56: $C_7\times D_4$, $C_7:D_4$, $D_{28}$, $C_4\times D_7$, $C_2\times D_{14}$

Order 48: $S_3\times D_4$

Order 42: $C_{42}$, $D_{21}$, $C_3\times D_7$, $S_3\times C_7$

Order 28: $D_{14}$ x 3, $C_2\times C_{14}$, $C_{28}$, $C_7:C_4$

Order 24: $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_2\times D_6$, $C_3\times D_4$

Order 21: $C_{21}$

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 2, $D_7$ x 2

Order 12: $D_6$ x 3, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 8: $D_4$ x 3, $C_2^3$, $C_2\times C_4$

Order 7: $C_7$

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 336: $C_{12}:D_{14}$ ($C_1$)

Order 168: $S_3\times D_{14}$ ($C_2$) x 2, $C_{21}:D_4$ ($C_2$) x 2, $C_4\times D_{21}$ ($C_2$), $C_7\times D_{12}$ ($C_2$), $C_3\times D_{28}$ ($C_2$)

Order 84: $S_3\times C_{14}$ ($C_2^2$) x 2, $C_3\times D_{14}$ ($C_2^2$) x 2, $C_{84}$ ($C_2^2$), $C_{21}:C_4$ ($C_2^2$), $D_{42}$ ($C_2^2$)

Order 56: $D_{28}$ ($S_3$)

Order 42: $D_{21}$ ($D_4$) x 2, $C_{42}$ ($C_2^3$)

Order 28: $D_{14}$ ($D_6$) x 2, $C_{28}$ ($D_6$)

Order 24: $D_{12}$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times D_4$)

Order 14: $C_{14}$ ($C_2\times D_6$)

Order 12: $D_6$ ($D_{14}$) x 2, $C_{12}$ ($D_{14}$)

Order 7: $C_7$ ($S_3\times D_4$)

Order 6: $C_6$ ($C_2\times D_{14}$)

Order 4: $C_4$ ($S_3\times D_7$)

Order 3: $C_3$ ($D_4\times D_7$)

Order 2: $C_2$ ($S_3\times D_{14}$)

Order 1: $C_1$ ($C_{12}:D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $C_{12}:D_{14}$ ($C_1$)

Order 168: $S_3\times D_{14}$ ($C_2$), $C_4\times D_{21}$ ($C_2$), $C_7\times D_{12}$ ($C_2$), $C_3\times D_{28}$ ($C_2$), $C_{21}:D_4$ ($C_2$)

Order 84: $C_{84}$ ($C_2^2$), $C_{21}:C_4$ ($C_2^2$), $D_{42}$ ($C_2^2$), $S_3\times C_{14}$ ($C_2^2$), $C_3\times D_{14}$ ($C_2^2$)

Order 56: $D_{28}$ ($S_3$)

Order 42: $C_{42}$ ($C_2^3$), $D_{21}$ ($D_4$)

Order 28: $D_{14}$ ($D_6$), $C_{28}$ ($D_6$)

Order 24: $D_{12}$ ($D_7$)

Order 21: $C_{21}$ ($C_2\times D_4$)

Order 14: $C_{14}$ ($C_2\times D_6$)

Order 12: $D_6$ ($D_{14}$), $C_{12}$ ($D_{14}$)

Order 7: $C_7$ ($S_3\times D_4$)

Order 6: $C_6$ ($C_2\times D_{14}$)

Order 4: $C_4$ ($S_3\times D_7$)

Order 3: $C_3$ ($D_4\times D_7$)

Order 2: $C_2$ ($S_3\times D_{14}$)

Order 1: $C_1$ ($C_{12}:D_{14}$)

Series

Derived series

$C_{12}:D_{14}$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_{12}:D_{14}$

$\rhd$

$S_3\times D_{14}$

$\rhd$

$S_3\times C_{14}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{12}:D_{14}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 44 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.