Abstract group 1344.2944: $D_{24}:D_{14}$

• Label: 1344.2944
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$D_{24}:D_{14}$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{84}.(C_2\times C_6).C_2^5$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
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	16	21	24	28	42	56	84	112	168
Elements	1	103	2	184	158	6	32	32	198	192	12	64	12	108	24	24	144	48	1344
Conjugacy classes	1	5	1	3	7	3	3	2	9	4	3	4	3	9	6	3	12	6	84
Divisions	1	5	1	3	4	1	2	2	3	2	1	2	1	2	1	1	1	1	34
Autjugacy classes	1	5	1	3	4	1	2	2	3	2	1	2	1	2	1	1	1	1	34

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	8	26	41	0	9	0	0	0	0	84
Irr. rational chars.	8	6	6	4	1	3	1	3	2	34

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	96
Arbitrary	6	8	16

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{112}=d^{3}=[a,b]=[a,d]=[b,d]=1, c^{a}=c^{41}, c^{b}=c^{15}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(11,12)(14,17)(15,18)(22,26), (9,10)(11,13)(12,16)(14,19) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{24}$ $\,\rtimes\,$ $D_{14}$	$(D_7\times D_8)$ $\,\rtimes\,$ $S_3$	$(D_{24}:D_7)$ $\,\rtimes\,$ $C_2$	$(C_3:D_{16})$ $\,\rtimes\,$ $D_7$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{56}$ . $D_6$	$C_{168}$ . $C_2^3$	$D_8$ . $(S_3\times D_7)$	$(C_7\times D_8)$ . $D_6$	all 28

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

Homology

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

Subgroups

There are 1860 subgroups in 180 conjugacy classes, 48 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{12}:D_{14}$
Commutator:	$G' \simeq$ $C_{168}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $D_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{24}:D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{16}:C_2$
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

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

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Classes of subgroups up to conjugation

Order 1344: $D_{24}:D_{14}$

Order 672: $C_{24}:D_{14}$, $C_{21}:\SD_{32}$, $C_{21}:D_{16}$, $C_{21}:\OD_{32}$, $D_{24}:D_7$, $C_{21}:\SD_{32}$, $C_{21}:D_{16}$

Order 448: $C_{16}:D_{14}$

Order 336: $C_{21}:Q_{16}$, $C_7\times D_{24}$, $C_{21}:D_8$, $C_3\times D_{56}$, $D_7\times C_{24}$, $C_{21}:C_{16}$, $C_{21}:\SD_{16}$, $C_3:C_{112}$, $C_{12}:D_{14}$, $D_{12}:D_7$, $D_8\times C_{21}$

Order 224: $C_7\times D_{16}$, $C_{112}:C_2$, $C_{112}:C_2$, $C_7:\SD_{32}$, $C_7:D_{16}$, $D_8:D_7$, $D_7\times D_8$

Order 192: $D_{24}:C_2^2$

Order 168: $C_{168}$, $C_6\times D_{14}$, $D_4\times C_{21}$, $C_7:C_{24}$, $C_{21}:Q_8$, $C_7\times D_{12}$, $C_{21}:D_4$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_{21}:D_4$, $C_6.D_{14}$

Order 112: $C_7\times D_8$ x 2, $C_7:Q_{16}$, $D_{56}$, $D_4:D_7$, $D_4\times D_7$, $C_8\times D_7$, $C_{112}$, $C_7:\SD_{16}$, $C_7:D_8$, $C_7:C_{16}$

Order 96: $C_3:\SD_{32}$ x 2, $C_3:D_{16}$ x 2, $C_3:\OD_{32}$, $C_6\times D_8$, $D_{24}:C_2$

Order 84: $C_3\times D_{14}$ x 4, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_2\times C_{42}$, $S_3\times C_{14}$

Order 64: $D_{16}:C_2$

Order 56: $C_7\times D_4$ x 2, $C_7:D_4$ x 2, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_{56}$, $C_2\times D_{14}$, $C_7:C_8$

Order 48: $C_3\times D_8$ x 3, $C_3:C_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $C_{24}:C_2$, $C_6\times D_4$, $D_{12}:C_2$, $C_2\times C_{24}$

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

Order 32: $\SD_{32}$ x 2, $D_{16}$ x 2, $D_8:C_2$, $C_2\times D_8$, $\OD_{32}$

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

Order 24: $C_3\times D_4$ x 3, $C_{24}$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $D_8$ x 4, $C_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $C_2\times C_8$, $D_4:C_2$, $C_2\times D_4$

Order 14: $C_{14}$ x 3, $D_7$ x 2

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 2, $D_6$, $C_3:C_4$

Order 8: $D_4$ x 5, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$, $Q_8$

Order 7: $C_7$

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

Order 4: $C_2^2$ x 6, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $D_{24}:D_{14}$

Order 672: $C_{24}:D_{14}$, $C_{21}:\SD_{32}$, $C_{21}:D_{16}$, $C_{21}:\OD_{32}$, $D_{24}:D_7$, $C_{21}:\SD_{32}$, $C_{21}:D_{16}$

Order 448: $C_{16}:D_{14}$

Order 336: $C_{21}:Q_{16}$, $C_7\times D_{24}$, $C_{21}:D_8$, $C_3\times D_{56}$, $D_7\times C_{24}$, $C_{21}:C_{16}$, $C_{21}:\SD_{16}$, $C_3:C_{112}$, $C_{12}:D_{14}$, $D_{12}:D_7$, $D_8\times C_{21}$

Order 224: $C_7\times D_{16}$, $C_{112}:C_2$, $C_{112}:C_2$, $C_7:\SD_{32}$, $C_7:D_{16}$, $D_8:D_7$, $D_7\times D_8$

Order 192: $D_{24}:C_2^2$

Order 168: $C_{168}$, $C_6\times D_{14}$, $D_4\times C_{21}$, $C_7:C_{24}$, $C_{21}:Q_8$, $C_7\times D_{12}$, $C_{21}:D_4$, $C_3\times D_{28}$, $C_{12}\times D_7$, $C_{21}:D_4$, $C_6.D_{14}$

Order 112: $C_7\times D_8$ x 2, $C_7:Q_{16}$, $D_{56}$, $D_4:D_7$, $D_4\times D_7$, $C_8\times D_7$, $C_{112}$, $C_7:\SD_{16}$, $C_7:D_8$, $C_7:C_{16}$

Order 96: $C_3:\SD_{32}$ x 2, $C_3:D_{16}$ x 2, $C_3:\OD_{32}$, $C_6\times D_8$, $D_{24}:C_2$

Order 84: $C_3\times D_{14}$ x 3, $C_{84}$, $C_{21}:C_4$, $C_7:C_{12}$, $C_2\times C_{42}$, $S_3\times C_{14}$

Order 64: $D_{16}:C_2$

Order 56: $C_7\times D_4$ x 2, $C_7:D_4$ x 2, $C_{14}:C_4$, $D_{28}$, $C_4\times D_7$, $C_7:Q_8$, $C_{56}$, $C_2\times D_{14}$, $C_7:C_8$

Order 48: $C_3\times D_8$ x 3, $C_3:C_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $C_{24}:C_2$, $C_6\times D_4$, $D_{12}:C_2$, $C_2\times C_{24}$

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

Order 32: $\SD_{32}$ x 2, $D_{16}$ x 2, $D_8:C_2$, $C_2\times D_8$, $\OD_{32}$

Order 28: $D_{14}$ x 3, $C_2\times C_{14}$ x 2, $C_7:C_4$ x 2, $C_{28}$

Order 24: $C_3\times D_4$ x 3, $C_{24}$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $D_8$ x 4, $C_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $C_2\times C_8$, $D_4:C_2$, $C_2\times D_4$

Order 14: $C_{14}$ x 3, $D_7$ x 2

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 2, $D_6$, $C_3:C_4$

Order 8: $D_4$ x 5, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$, $Q_8$

Order 7: $C_7$

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

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $D_{24}:D_{14}$ ($C_1$)

Order 672: $C_{24}:D_{14}$ ($C_2$), $C_{21}:\SD_{32}$ ($C_2$), $C_{21}:D_{16}$ ($C_2$), $C_{21}:\OD_{32}$ ($C_2$), $D_{24}:D_7$ ($C_2$), $C_{21}:\SD_{32}$ ($C_2$), $C_{21}:D_{16}$ ($C_2$)

Order 336: $C_{21}:Q_{16}$ ($C_2^2$), $C_7\times D_{24}$ ($C_2^2$), $C_3\times D_{56}$ ($C_2^2$), $D_7\times C_{24}$ ($C_2^2$), $C_{21}:C_{16}$ ($C_2^2$), $C_3:C_{112}$ ($C_2^2$), $D_8\times C_{21}$ ($C_2^2$)

Order 224: $D_7\times D_8$ ($S_3$)

Order 168: $C_{168}$ ($C_2^3$), $C_7:C_{24}$ ($D_4$), $C_{12}\times D_7$ ($D_4$)

Order 112: $D_{56}$ ($D_6$), $C_8\times D_7$ ($D_6$), $C_7\times D_8$ ($D_6$)

Order 96: $C_3:D_{16}$ ($D_7$)

Order 84: $C_{84}$ ($C_2\times D_4$), $C_7:C_{12}$ ($D_8$), $C_3\times D_{14}$ ($D_8$)

Order 56: $C_4\times D_7$ ($C_3:D_4$), $C_{56}$ ($C_2\times D_6$), $C_7:C_8$ ($C_3:D_4$)

Order 48: $D_{24}$ ($D_{14}$), $C_3\times D_8$ ($D_{14}$), $C_3:C_{16}$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2\times D_8$)

Order 28: $D_{14}$ ($C_3:D_8$), $C_{28}$ ($C_6:D_4$), $C_7:C_4$ ($C_3:D_8$)

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

Order 21: $C_{21}$ ($D_{16}:C_2$)

Order 16: $D_8$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_6:D_8$)

Order 12: $C_{12}$ ($D_4\times D_7$)

Order 8: $C_8$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($D_{24}:C_2^2$)

Order 6: $C_6$ ($D_7\times D_8$)

Order 4: $C_4$ ($D_6:D_{14}$)

Order 3: $C_3$ ($C_{16}:D_{14}$)

Order 2: $C_2$ ($D_{12}:D_{14}$)

Order 1: $C_1$ ($D_{24}:D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $D_{24}:D_{14}$ ($C_1$)

Order 672: $C_{24}:D_{14}$ ($C_2$), $C_{21}:\SD_{32}$ ($C_2$), $C_{21}:D_{16}$ ($C_2$), $C_{21}:\OD_{32}$ ($C_2$), $D_{24}:D_7$ ($C_2$), $C_{21}:\SD_{32}$ ($C_2$), $C_{21}:D_{16}$ ($C_2$)

Order 336: $C_{21}:Q_{16}$ ($C_2^2$), $C_7\times D_{24}$ ($C_2^2$), $C_3\times D_{56}$ ($C_2^2$), $D_7\times C_{24}$ ($C_2^2$), $C_{21}:C_{16}$ ($C_2^2$), $C_3:C_{112}$ ($C_2^2$), $D_8\times C_{21}$ ($C_2^2$)

Order 224: $D_7\times D_8$ ($S_3$)

Order 168: $C_{168}$ ($C_2^3$), $C_7:C_{24}$ ($D_4$), $C_{12}\times D_7$ ($D_4$)

Order 112: $D_{56}$ ($D_6$), $C_8\times D_7$ ($D_6$), $C_7\times D_8$ ($D_6$)

Order 96: $C_3:D_{16}$ ($D_7$)

Order 84: $C_{84}$ ($C_2\times D_4$), $C_7:C_{12}$ ($D_8$), $C_3\times D_{14}$ ($D_8$)

Order 56: $C_4\times D_7$ ($C_3:D_4$), $C_{56}$ ($C_2\times D_6$), $C_7:C_8$ ($C_3:D_4$)

Order 48: $D_{24}$ ($D_{14}$), $C_3\times D_8$ ($D_{14}$), $C_3:C_{16}$ ($D_{14}$)

Order 42: $C_{42}$ ($C_2\times D_8$)

Order 28: $D_{14}$ ($C_3:D_8$), $C_{28}$ ($C_6:D_4$), $C_7:C_4$ ($C_3:D_8$)

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

Order 21: $C_{21}$ ($D_{16}:C_2$)

Order 16: $D_8$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($C_6:D_8$)

Order 12: $C_{12}$ ($D_4\times D_7$)

Order 8: $C_8$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($D_{24}:C_2^2$)

Order 6: $C_6$ ($D_7\times D_8$)

Order 4: $C_4$ ($D_6:D_{14}$)

Order 3: $C_3$ ($C_{16}:D_{14}$)

Order 2: $C_2$ ($D_{12}:D_{14}$)

Order 1: $C_1$ ($D_{24}:D_{14}$)

Series

Derived series

$D_{24}:D_{14}$

$\rhd$

$C_{168}$

$\rhd$

$C_1$

Chief series

$D_{24}:D_{14}$

$\rhd$

$C_{24}:D_{14}$

$\rhd$

$D_7\times C_{24}$

$\rhd$

$C_{168}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{24}:D_{14}$

$\rhd$

$C_{168}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

$\lhd$

$D_8$

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.