Abstract group 672.809: $C_{24}.D_{14}$

• Label: 672.809
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $3$

Group information

Description:	$C_{24}.D_{14}$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$D_4\times C_2^3\times F_7$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$
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	24	28	42	56	84	168
Elements	1	47	2	48	94	6	32	96	30	12	64	36	60	24	72	48	672
Conjugacy classes	1	4	2	5	8	3	4	10	6	6	8	6	12	6	12	12	105
Divisions	1	4	1	4	4	1	2	4	2	1	2	2	2	1	2	1	34
Autjugacy classes	1	4	1	4	4	1	2	4	2	1	2	2	2	1	2	1	34

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	24	54	27	0	0	0	0	0	0	105
Irr. rational chars.	8	10	2	4	1	5	1	2	1	34

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$3$
Inequivalent generating triples:	$34944$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	48
Arbitrary	4	6	16

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{24}=[a,c]=1, b^{14}=c^{12}, b^{a}=b^{13}c^{12}, c^{b}=c^{19} \rangle$
Permutation group:	Degree $26$ $\langle(2,3)(4,5)(6,7)(12,16)(13,19)(18,22)(21,23), (11,12)(13,20)(14,22)(15,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 92 & 77 \\ 70 & 78 \end{array}\right), \left(\begin{array}{rr} 18 & 0 \\ 0 & 18 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 69 & 0 \\ 91 & 118 \end{array}\right), \left(\begin{array}{rr} 50 & 0 \\ 0 & 50 \end{array}\right), \left(\begin{array}{rr} 64 & 28 \\ 112 & 113 \end{array}\right), \left(\begin{array}{rr} 57 & 100 \\ 112 & 62 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/119\Z)$
Direct product:	$C_3$ $\, \times\, $ $(C_8.D_{14})$
Semidirect product:	$(D_4:D_7)$ $\,\rtimes\,$ $C_6$	$(C_7:D_8)$ $\,\rtimes\,$ $C_6$	$(C_8\times D_7)$ $\,\rtimes\,$ $C_6$	$C_7$ $\,\rtimes\,$ $(D_8:C_6)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{24}$ . $D_{14}$	$C_{84}$ . $C_2^3$	$C_{168}$ . $C_2^2$	$C_6$ . $(D_4\times D_7)$	all 32

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 556 subgroups in 124 conjugacy classes, 54 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_4\times D_7$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_6\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $\SD_{16}\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{24}.D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
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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Classes of subgroups up to conjugation

Order 672: $C_{24}.D_{14}$

Order 336: $C_{21}:Q_{16}$, $C_{21}:D_8$, $C_{168}:C_2$, $D_7\times C_{24}$, $D_{28}:C_6$, $C_{12}.D_{14}$, $\SD_{16}\times C_{21}$

Order 224: $C_8.D_{14}$

Order 168: $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_{168}$, $Q_8\times C_{21}$, $D_4\times C_{21}$, $C_7:C_{24}$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_{21}:Q_8$

Order 112: $C_{56}:C_2$, $D_{28}:C_2$, $D_4:D_7$, $C_8\times D_7$, $C_7\times \SD_{16}$, $C_7:Q_{16}$, $C_7:D_8$

Order 96: $D_8:C_6$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_3\times D_{14}$ x 2, $C_2\times C_{42}$

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

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

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

Order 32: $D_8:C_2$

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

Order 24: $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $C_{24}$ x 2, $C_3\times Q_8$ x 2

Order 21: $C_{21}$

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

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3

Order 8: $D_4$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $C_{24}.D_{14}$

Order 336: $C_{21}:Q_{16}$, $C_{21}:D_8$, $C_{168}:C_2$, $D_7\times C_{24}$, $D_{28}:C_6$, $C_{12}.D_{14}$, $\SD_{16}\times C_{21}$

Order 224: $C_8.D_{14}$

Order 168: $C_3\times D_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_{168}$, $Q_8\times C_{21}$, $D_4\times C_{21}$, $C_7:C_{24}$, $C_{21}:D_4$, $C_{14}:C_{12}$, $C_{21}:Q_8$

Order 112: $C_{56}:C_2$, $D_{28}:C_2$, $D_4:D_7$, $C_8\times D_7$, $C_7\times \SD_{16}$, $C_7:Q_{16}$, $C_7:D_8$

Order 96: $D_8:C_6$

Order 84: $C_{84}$ x 2, $C_7:C_{12}$ x 2, $C_3\times D_{14}$ x 2, $C_2\times C_{42}$

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

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

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

Order 32: $D_8:C_2$

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

Order 24: $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $C_{24}$ x 2, $C_3\times Q_8$ x 2

Order 21: $C_{21}$

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

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

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3

Order 8: $D_4$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $C_{24}.D_{14}$ ($C_1$)

Order 336: $C_{21}:Q_{16}$ ($C_2$), $C_{21}:D_8$ ($C_2$), $C_{168}:C_2$ ($C_2$), $D_7\times C_{24}$ ($C_2$), $D_{28}:C_6$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $\SD_{16}\times C_{21}$ ($C_2$)

Order 224: $C_8.D_{14}$ ($C_3$)

Order 168: $C_{168}$ ($C_2^2$), $Q_8\times C_{21}$ ($C_2^2$), $D_4\times C_{21}$ ($C_2^2$), $C_7:C_{24}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$)

Order 112: $C_{56}:C_2$ ($C_6$), $D_{28}:C_2$ ($C_6$), $D_4:D_7$ ($C_6$), $C_8\times D_7$ ($C_6$), $C_7\times \SD_{16}$ ($C_6$), $C_7:Q_{16}$ ($C_6$), $C_7:D_8$ ($C_6$)

Order 84: $C_{84}$ ($C_2^3$), $C_7:C_{12}$ ($D_4$), $C_3\times D_{14}$ ($D_4$)

Order 56: $C_7\times D_4$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_4\times D_7$ ($C_2\times C_6$), $C_7:Q_8$ ($C_2\times C_6$), $C_{56}$ ($C_2\times C_6$), $C_7\times Q_8$ ($C_2\times C_6$), $C_7:C_8$ ($C_2\times C_6$)

Order 48: $C_3\times \SD_{16}$ ($D_7$)

Order 42: $C_{42}$ ($C_2\times D_4$)

Order 28: $D_{14}$ ($C_3\times D_4$), $C_{28}$ ($C_2^2\times C_6$), $C_7:C_4$ ($C_3\times D_4$)

Order 24: $C_{24}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$), $C_3\times D_4$ ($D_{14}$)

Order 21: $C_{21}$ ($D_8:C_2$)

Order 16: $\SD_{16}$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_6\times D_4$)

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

Order 8: $Q_8$ ($C_3\times D_{14}$), $D_4$ ($C_3\times D_{14}$), $C_8$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($D_8:C_6$)

Order 6: $C_6$ ($D_4\times D_7$)

Order 4: $C_4$ ($C_6\times D_{14}$)

Order 3: $C_3$ ($C_8.D_{14}$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $C_{24}.D_{14}$ ($C_1$)

Order 336: $C_{21}:Q_{16}$ ($C_2$), $C_{21}:D_8$ ($C_2$), $C_{168}:C_2$ ($C_2$), $D_7\times C_{24}$ ($C_2$), $D_{28}:C_6$ ($C_2$), $C_{12}.D_{14}$ ($C_2$), $\SD_{16}\times C_{21}$ ($C_2$)

Order 224: $C_8.D_{14}$ ($C_3$)

Order 168: $C_{168}$ ($C_2^2$), $Q_8\times C_{21}$ ($C_2^2$), $D_4\times C_{21}$ ($C_2^2$), $C_7:C_{24}$ ($C_2^2$), $C_3\times D_{28}$ ($C_2^2$), $C_{12}\times D_7$ ($C_2^2$), $C_{21}:Q_8$ ($C_2^2$)

Order 112: $C_{56}:C_2$ ($C_6$), $D_{28}:C_2$ ($C_6$), $D_4:D_7$ ($C_6$), $C_8\times D_7$ ($C_6$), $C_7\times \SD_{16}$ ($C_6$), $C_7:Q_{16}$ ($C_6$), $C_7:D_8$ ($C_6$)

Order 84: $C_{84}$ ($C_2^3$), $C_7:C_{12}$ ($D_4$), $C_3\times D_{14}$ ($D_4$)

Order 56: $C_7\times D_4$ ($C_2\times C_6$), $D_{28}$ ($C_2\times C_6$), $C_4\times D_7$ ($C_2\times C_6$), $C_7:Q_8$ ($C_2\times C_6$), $C_{56}$ ($C_2\times C_6$), $C_7\times Q_8$ ($C_2\times C_6$), $C_7:C_8$ ($C_2\times C_6$)

Order 48: $C_3\times \SD_{16}$ ($D_7$)

Order 42: $C_{42}$ ($C_2\times D_4$)

Order 28: $D_{14}$ ($C_3\times D_4$), $C_{28}$ ($C_2^2\times C_6$), $C_7:C_4$ ($C_3\times D_4$)

Order 24: $C_{24}$ ($D_{14}$), $C_3\times Q_8$ ($D_{14}$), $C_3\times D_4$ ($D_{14}$)

Order 21: $C_{21}$ ($D_8:C_2$)

Order 16: $\SD_{16}$ ($C_3\times D_7$)

Order 14: $C_{14}$ ($C_6\times D_4$)

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

Order 8: $Q_8$ ($C_3\times D_{14}$), $D_4$ ($C_3\times D_{14}$), $C_8$ ($C_3\times D_{14}$)

Order 7: $C_7$ ($D_8:C_6$)

Order 6: $C_6$ ($D_4\times D_7$)

Order 4: $C_4$ ($C_6\times D_{14}$)

Order 3: $C_3$ ($C_8.D_{14}$)

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

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

Series

Derived series

$C_{24}.D_{14}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$C_{24}.D_{14}$

$\rhd$

$D_7\times C_{24}$

$\rhd$

$C_{168}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}.D_{14}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times \SD_{16}$

Supergroups

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

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

Character theory

Complex character table

See the $105 \times 105$ 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.