Abstract group 1344.1752: $C_{12}.(C_4\times D_{14})$

• Label: 1344.1752
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $1344$
• Rank: $3$

Group information

Description:	$C_{12}.(C_4\times D_{14})$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{42}.(C_2^5\times C_6).C_2^3$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
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	8	12	14	21	28	42	84
Elements	1	3	2	172	6	6	336	248	18	12	360	36	144	1344
Conjugacy classes	1	3	1	14	3	3	4	16	9	3	48	9	36	150
Divisions	1	3	1	10	3	1	3	7	3	1	7	3	4	47
Autjugacy classes	1	3	1	8	3	1	2	6	3	1	6	3	4	42

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	16	68	66	0	0	0	0	0	150
Irr. rational chars.	8	10	7	4	3	8	5	2	47

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$1344$
Rank:	$3$
Inequivalent generating triples:	$10752$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	10	16

Constructions

Presentation:	$\langle a, b, c, d \mid b^{4}=c^{8}=d^{21}=[a,b]=[b,d]=1, a^{2}=c^{4}, c^{a}=b^{2}c^{3}, d^{a}=d^{8}, c^{b}=c^{5}, d^{c}=d^{20} \rangle$
Permutation group:	Degree $30$ $\langle(2,3)(4,5)(6,7)(11,12,15,19)(13,20,22,26)(14,18,16,24)(17,21,25,23)(27,28) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{21}:Q_{16})$ $\,\rtimes\,$ $C_4$ (4)	$C_{21}$ $\,\rtimes\,$ $(Q_{16}:C_4)$	$C_7$ $\,\rtimes\,$ $(C_4^2.D_6)$	$C_3$ $\,\rtimes\,$ $(C_4^2.D_{14})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{28}:Q_8)$ . $S_3$	$(C_{14}:Q_8)$ . $D_6$	$(C_{12}:Q_8)$ . $D_7$	$(C_6:Q_8)$ . $D_{14}$	all 58

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

Homology

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

Subgroups

There are 1124 subgroups in 216 conjugacy classes, 78 normal (64 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_{42}:D_4$
Commutator:	$G' \simeq$ $C_{84}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}.(C_4\times D_{14})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}:C_4$
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

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

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.

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

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

Classes of subgroups up to conjugation

Order 1344: $C_{12}.(C_4\times D_{14})$

Order 672: $C_{28}.D_{12}$, $C_{12}.D_{28}$, $C_{84}:Q_8$, $C_{84}:Q_8$, $C_{42}:Q_{16}$, $C_{42}.C_4^2$, $C_{84}.Q_8$

Order 448: $C_4^2.D_{14}$

Order 336: $C_{21}:Q_{16}$ x 4, $C_{42}:C_8$ x 2, $C_{12}:C_{28}$, $C_{42}.D_4$, $C_{12}:C_{28}$, $C_{28}:C_{12}$, $C_{42}.D_4$, $C_{28}:C_{12}$, $C_{42}:Q_8$, $C_{42}:Q_8$, $C_4\times C_{84}$

Order 224: $C_{14}.C_4^2$, $C_{28}:Q_8$, $C_{14}.Q_{16}$, $C_4.D_{28}$, $Q_8\times C_{28}$, $C_{28}.Q_8$, $C_{14}:Q_{16}$

Order 192: $C_4^2.D_6$

Order 168: $C_{21}:C_8$ x 3, $C_2\times C_{84}$ x 3, $C_{21}:Q_8$ x 3, $C_{21}:Q_8$ x 3, $C_6:C_{28}$ x 2, $C_{14}:C_{12}$ x 2

Order 112: $C_7:Q_{16}$ x 4, $C_{14}:C_8$ x 2, $C_4:C_{28}$ x 2, $C_4\times C_{28}$ x 2, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_{28}:C_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_{12}:Q_8$, $C_6.Q_{16}$, $C_4.D_{12}$, $Q_8\times C_{12}$, $C_6:Q_{16}$, $C_{12}.Q_8$, $C_6.C_4^2$

Order 84: $C_{84}$ x 4, $C_7:C_{12}$ x 3, $C_3:C_{28}$ x 3, $C_2\times C_{42}$

Order 64: $Q_{16}:C_4$

Order 56: $C_2\times C_{28}$ x 5, $C_7:Q_8$ x 3, $C_7\times Q_8$ x 3, $C_7:C_8$ x 3, $C_{14}:C_4$ x 2

Order 48: $C_3:Q_{16}$ x 4, $C_6:C_8$ x 2, $C_4:C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3

Order 32: $C_4\times Q_8$ x 2, $Q_8:C_4$ x 2, $C_2\times Q_{16}$, $C_8:C_4$, $C_8:C_4$

Order 28: $C_{28}$ x 7, $C_7:C_4$ x 3, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 5, $C_3:Q_8$ x 3, $C_3\times Q_8$ x 3, $C_3:C_8$ x 3, $C_6:C_4$ x 2

Order 21: $C_{21}$

Order 16: $Q_{16}$ x 4, $C_4:C_4$ x 4, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 7, $C_3:C_4$ x 3, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 6, $C_8$ x 3

Order 7: $C_7$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 10, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{12}.(C_4\times D_{14})$

Order 672: $C_{28}.D_{12}$, $C_{12}.D_{28}$, $C_{84}:Q_8$, $C_{84}:Q_8$, $C_{42}:Q_{16}$, $C_{42}.C_4^2$, $C_{84}.Q_8$

Order 448: $C_4^2.D_{14}$

Order 336: $C_{42}:C_8$ x 2, $C_{12}:C_{28}$, $C_{42}.D_4$, $C_{12}:C_{28}$, $C_{28}:C_{12}$, $C_{42}.D_4$, $C_{28}:C_{12}$, $C_{21}:Q_{16}$, $C_{42}:Q_8$, $C_{42}:Q_8$, $C_4\times C_{84}$

Order 224: $C_{14}.C_4^2$, $C_{28}:Q_8$, $C_{14}.Q_{16}$, $C_4.D_{28}$, $Q_8\times C_{28}$, $C_{28}.Q_8$, $C_{14}:Q_{16}$

Order 192: $C_4^2.D_6$

Order 168: $C_2\times C_{84}$ x 3, $C_{21}:C_8$ x 2, $C_6:C_{28}$ x 2, $C_{21}:Q_8$ x 2, $C_{14}:C_{12}$ x 2, $C_{21}:Q_8$ x 2

Order 112: $C_{14}:C_8$ x 2, $C_4:C_{28}$ x 2, $C_4\times C_{28}$ x 2, $Q_8\times C_{14}$, $C_{14}:Q_8$, $C_7:Q_{16}$, $C_{28}:C_4$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $C_{12}:Q_8$, $C_6.Q_{16}$, $C_4.D_{12}$, $Q_8\times C_{12}$, $C_6:Q_{16}$, $C_{12}.Q_8$, $C_6.C_4^2$

Order 84: $C_{84}$ x 4, $C_7:C_{12}$ x 2, $C_3:C_{28}$ x 2, $C_2\times C_{42}$

Order 64: $Q_{16}:C_4$

Order 56: $C_2\times C_{28}$ x 5, $C_{14}:C_4$ x 2, $C_7:Q_8$ x 2, $C_7\times Q_8$ x 2, $C_7:C_8$ x 2

Order 48: $C_6:C_8$ x 2, $C_4:C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_3:Q_{16}$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3

Order 32: $C_4\times Q_8$ x 2, $Q_8:C_4$ x 2, $C_2\times Q_{16}$, $C_8:C_4$, $C_8:C_4$

Order 28: $C_{28}$ x 6, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 5, $C_6:C_4$ x 2, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 4, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2, $Q_{16}$

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 6, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 8, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{12}.(C_4\times D_{14})$ ($C_1$)

Order 672: $C_{28}.D_{12}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $C_{42}:Q_{16}$ ($C_2$), $C_{42}.C_4^2$ ($C_2$), $C_{84}.Q_8$ ($C_2$)

Order 336: $C_{21}:Q_{16}$ ($C_4$) x 4, $C_{42}:C_8$ ($C_2^2$) x 2, $C_{12}:C_{28}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_4\times C_{84}$ ($C_2^2$)

Order 224: $C_{28}:Q_8$ ($S_3$)

Order 168: $C_{21}:C_8$ ($C_2\times C_4$) x 2, $C_2\times C_{84}$ ($D_4$) x 2, $C_{21}:Q_8$ ($C_2\times C_4$) x 2, $C_{21}:Q_8$ ($C_2\times C_4$) x 2, $C_2\times C_{84}$ ($C_2^3$)

Order 112: $C_{14}:Q_8$ ($D_6$), $C_4\times C_{28}$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $C_{12}:Q_8$ ($D_7$)

Order 84: $C_{84}$ ($D_4:C_2$), $C_{84}$ ($C_2^2\times C_4$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($C_3:D_4$) x 2, $C_7:Q_8$ ($C_4\times S_3$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$)

Order 48: $C_6:Q_8$ ($D_{14}$), $C_4\times C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($Q_{16}:C_2$) x 2, $C_{42}$ ($C_4\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_6:D_4$), $C_{28}$ ($D_{12}:C_2$), $C_{28}$ ($C_4\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_7:D_4$) x 2, $C_3:Q_8$ ($C_4\times D_7$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($Q_{16}:C_4$)

Order 16: $C_4^2$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_{12}.D_4$), $C_{14}$ ($C_{12}.D_4$), $C_{14}$ ($C_{12}:D_4$)

Order 12: $C_2\times C_6$ ($C_{14}:D_4$), $C_{12}$ ($D_{28}:C_2$), $C_{12}$ ($C_4\times D_{14}$)

Order 8: $C_2\times C_4$ ($C_{21}:D_4$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_4^2.D_6$)

Order 6: $C_6$ ($C_{28}.D_4$), $C_6$ ($C_{28}.D_4$), $C_6$ ($C_{28}:D_4$)

Order 4: $C_2^2$ ($C_{42}:D_4$), $C_4$ ($C_{12}:D_{14}$), $C_4$ ($C_{12}.D_{14}$)

Order 3: $C_3$ ($C_4^2.D_{14}$)

Order 2: $C_2$ ($C_{84}:D_4$), $C_2$ ($C_{84}.D_4$), $C_2$ ($C_{84}.D_4$)

Order 1: $C_1$ ($C_{12}.(C_4\times D_{14})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{12}.(C_4\times D_{14})$ ($C_1$)

Order 672: $C_{28}.D_{12}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $C_{84}:Q_8$ ($C_2$), $C_{42}:Q_{16}$ ($C_2$), $C_{42}.C_4^2$ ($C_2$), $C_{84}.Q_8$ ($C_2$)

Order 336: $C_{42}:C_8$ ($C_2^2$) x 2, $C_{12}:C_{28}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $C_{21}:Q_{16}$ ($C_4$), $C_{42}:Q_8$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_4\times C_{84}$ ($C_2^2$)

Order 224: $C_{28}:Q_8$ ($S_3$)

Order 168: $C_2\times C_{84}$ ($D_4$) x 2, $C_{21}:C_8$ ($C_2\times C_4$), $C_2\times C_{84}$ ($C_2^3$), $C_{21}:Q_8$ ($C_2\times C_4$), $C_{21}:Q_8$ ($C_2\times C_4$)

Order 112: $C_{14}:Q_8$ ($D_6$), $C_4\times C_{28}$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $C_{12}:Q_8$ ($D_7$)

Order 84: $C_{84}$ ($D_4:C_2$), $C_{84}$ ($C_2^2\times C_4$), $C_2\times C_{42}$ ($C_2\times D_4$)

Order 56: $C_2\times C_{28}$ ($C_3:D_4$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $C_7:Q_8$ ($C_4\times S_3$)

Order 48: $C_6:Q_8$ ($D_{14}$), $C_4\times C_{12}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($Q_{16}:C_2$) x 2, $C_{42}$ ($C_4\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_6:D_4$), $C_{28}$ ($D_{12}:C_2$), $C_{28}$ ($C_4\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_7:D_4$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_3:Q_8$ ($C_4\times D_7$)

Order 21: $C_{21}$ ($Q_{16}:C_4$)

Order 16: $C_4^2$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_{12}.D_4$), $C_{14}$ ($C_{12}.D_4$), $C_{14}$ ($C_{12}:D_4$)

Order 12: $C_2\times C_6$ ($C_{14}:D_4$), $C_{12}$ ($D_{28}:C_2$), $C_{12}$ ($C_4\times D_{14}$)

Order 8: $C_2\times C_4$ ($C_{21}:D_4$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_4^2.D_6$)

Order 6: $C_6$ ($C_{28}.D_4$), $C_6$ ($C_{28}.D_4$), $C_6$ ($C_{28}:D_4$)

Order 4: $C_2^2$ ($C_{42}:D_4$), $C_4$ ($C_{12}:D_{14}$), $C_4$ ($C_{12}.D_{14}$)

Order 3: $C_3$ ($C_4^2.D_{14}$)

Order 2: $C_2$ ($C_{84}:D_4$), $C_2$ ($C_{84}.D_4$), $C_2$ ($C_{84}.D_4$)

Order 1: $C_1$ ($C_{12}.(C_4\times D_{14})$)

Series

Derived series

$C_{12}.(C_4\times D_{14})$

$\rhd$

$C_{84}$

$\rhd$

$C_1$

Chief series

$C_{12}.(C_4\times D_{14})$

$\rhd$

$C_{84}:Q_8$

$\rhd$

$C_{42}:Q_8$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{12}.(C_4\times D_{14})$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4^2$

Character theory

Complex character table

See the $150 \times 150$ 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 $47 \times 47$ rational character table.