Abstract group 1344.2324: $(C_2\times C_{24}).D_{14}$

• Label: 1344.2324
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $34$
• Trans deg.: $672$
• Rank: $3$

Group information

Description:	$(C_2\times C_{24}).D_{14}$
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	24	28	42	56	84	168
Elements	1	15	2	464	6	6	32	232	90	12	16	96	36	192	48	96	1344
Conjugacy classes	1	4	1	10	3	3	4	6	15	3	4	18	9	24	12	24	141
Divisions	1	4	1	7	3	1	2	4	4	1	1	3	3	2	2	1	40
Autjugacy classes	1	4	1	7	3	1	2	4	4	1	1	3	3	2	2	1	40

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	8	66	67	0	0	0	0	0	141
Irr. rational chars.	8	6	5	4	4	6	4	3	40

Minimal presentations

Permutation degree:	$34$
Transitive degree:	$672$
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	4	8	18

Constructions

Presentation:	$\langle a, b, c \mid b^{4}=c^{168}=[a,b]=1, a^{2}=c^{84}, c^{a}=b^{2}c^{113}, c^{b}=c^{167} \rangle$
Permutation group:	Degree $34$ $\langle(2,3)(4,5)(6,7)(11,12,16,19)(13,17,15,24)(14,21,23,26)(18,20,25,22)(27,28,31,30) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{84}.Q_8)$ $\,\rtimes\,$ $C_2$	$(C_{84}.D_4)$ $\,\rtimes\,$ $C_2$	$(C_{84}.D_4)$ $\,\rtimes\,$ $C_2$	$(C_{14}.Q_{16})$ $\,\rtimes\,$ $S_3$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_6:C_8)$ . $D_7$	$C_{28}$ . $(D_4:S_3)$	$(C_{14}:Q_8)$ . $D_6$	$(D_6\times C_{14})$ . $D_4$	all 47

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

Homology

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

Subgroups

There are 1428 subgroups in 192 conjugacy classes, 60 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3\times D_{28}$
Commutator:	$G' \simeq$ $C_2\times C_{84}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{168}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(C_2\times C_{24}).D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_4$
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 1344: $(C_2\times C_{24}).D_{14}$

Order 672: $C_{84}.D_4$, $C_{84}.D_4$, $(S_3\times C_{28}):C_4$, $C_{168}:C_4$, $D_6:C_{56}$, $C_{84}.D_4$, $C_{84}.Q_8$

Order 448: $C_2^3.D_{28}$

Order 336: $C_{84}:C_4$ x 2, $C_{14}.D_{12}$ x 2, $C_6:C_{56}$, $C_{28}:C_{12}$, $C_{42}.D_4$, $C_{21}:C_4^2$, $D_6\times C_{28}$, $C_{42}:Q_8$, $C_2\times C_{168}$

Order 224: $C_{14}.Q_{16}$ x 2, $C_2^2:C_{56}$, $C_{56}:C_4$, $C_{56}:C_4$, $C_2^3.D_{14}$, $C_{28}.D_4$

Order 192: $(C_2\times C_8).D_6$

Order 168: $C_{42}:C_4$ x 2, $S_3\times C_{28}$ x 2, $C_{14}:C_{12}$ x 2, $C_{21}:Q_8$ x 2, $C_{168}$, $D_6\times C_{14}$, $C_2\times C_{84}$, $C_6:C_{28}$, $C_3:C_{56}$

Order 112: $C_{28}:C_4$ x 3, $C_2\times C_{56}$ x 2, $C_{14}.D_4$ x 2, $C_2^2\times C_{28}$, $C_{14}:Q_8$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $(C_4\times S_3):C_4$, $Q_8:C_{12}$, $C_6.Q_{16}$, $D_6:C_8$, $C_{24}:C_4$, $D_6:Q_8$, $C_{12}.Q_8$

Order 84: $S_3\times C_{14}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$ x 2, $C_7:C_{12}$ x 2, $C_3:C_{28}$, $C_2\times C_{42}$

Order 64: $C_2^3.D_4$

Order 56: $C_2\times C_{28}$ x 4, $C_{14}:C_4$ x 4, $C_7:Q_8$ x 2, $C_{56}$ x 2, $C_2^2\times C_{14}$

Order 48: $D_6:C_4$ x 2, $C_{12}:C_4$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_4\times D_6$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3, $S_3\times C_7$

Order 32: $Q_8:C_4$ x 2, $C_2^2:C_8$, $C_2^2:Q_8$, $C_4^2:C_2$, $C_8:C_4$, $C_8:C_4$

Order 28: $C_2\times C_{14}$ x 4, $C_7:C_4$ x 4, $C_{28}$ x 3

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$ x 2, $C_3\times Q_8$ x 2, $C_{24}$, $C_2\times D_6$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 4, $C_2\times C_8$ x 2, $C_2^2:C_4$ x 2, $C_4^2$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 14: $C_{14}$ x 4

Order 12: $C_{12}$ x 4, $D_6$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$

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

Order 7: $C_7$

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

Order 4: $C_4$ x 7, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $(C_2\times C_{24}).D_{14}$

Order 672: $C_{84}.D_4$, $C_{84}.D_4$, $(S_3\times C_{28}):C_4$, $C_{168}:C_4$, $D_6:C_{56}$, $C_{84}.D_4$, $C_{84}.Q_8$

Order 448: $C_2^3.D_{28}$

Order 336: $C_{84}:C_4$ x 2, $C_{14}.D_{12}$ x 2, $C_6:C_{56}$, $C_{28}:C_{12}$, $C_{42}.D_4$, $C_{21}:C_4^2$, $D_6\times C_{28}$, $C_{42}:Q_8$, $C_2\times C_{168}$

Order 224: $C_{14}.Q_{16}$ x 2, $C_2^2:C_{56}$, $C_{56}:C_4$, $C_{56}:C_4$, $C_2^3.D_{14}$, $C_{28}.D_4$

Order 192: $(C_2\times C_8).D_6$

Order 168: $C_{42}:C_4$ x 2, $S_3\times C_{28}$ x 2, $C_{14}:C_{12}$ x 2, $C_{21}:Q_8$ x 2, $C_{168}$, $D_6\times C_{14}$, $C_2\times C_{84}$, $C_6:C_{28}$, $C_3:C_{56}$

Order 112: $C_{28}:C_4$ x 3, $C_2\times C_{56}$ x 2, $C_{14}.D_4$ x 2, $C_2^2\times C_{28}$, $C_{14}:Q_8$, $C_{14}.D_4$, $C_{28}:C_4$

Order 96: $(C_4\times S_3):C_4$, $Q_8:C_{12}$, $C_6.Q_{16}$, $D_6:C_8$, $C_{24}:C_4$, $D_6:Q_8$, $C_{12}.Q_8$

Order 84: $S_3\times C_{14}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$ x 2, $C_7:C_{12}$ x 2, $C_3:C_{28}$, $C_2\times C_{42}$

Order 64: $C_2^3.D_4$

Order 56: $C_2\times C_{28}$ x 4, $C_{14}:C_4$ x 4, $C_7:Q_8$ x 2, $C_{56}$ x 2, $C_2^2\times C_{14}$

Order 48: $D_6:C_4$ x 2, $C_{12}:C_4$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_4\times D_6$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3, $S_3\times C_7$

Order 32: $Q_8:C_4$ x 2, $C_2^2:C_8$, $C_2^2:Q_8$, $C_4^2:C_2$, $C_8:C_4$, $C_8:C_4$

Order 28: $C_2\times C_{14}$ x 4, $C_7:C_4$ x 4, $C_{28}$ x 3

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$ x 2, $C_3\times Q_8$ x 2, $C_{24}$, $C_2\times D_6$, $C_3:C_8$

Order 21: $C_{21}$

Order 16: $C_4:C_4$ x 4, $C_2\times C_8$ x 2, $C_2^2:C_4$ x 2, $C_4^2$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 14: $C_{14}$ x 4

Order 12: $C_{12}$ x 4, $D_6$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$

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

Order 7: $C_7$

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

Order 4: $C_4$ x 7, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $(C_2\times C_{24}).D_{14}$ ($C_1$)

Order 672: $C_{84}.D_4$ ($C_2$), $C_{84}.D_4$ ($C_2$), $(S_3\times C_{28}):C_4$ ($C_2$), $C_{168}:C_4$ ($C_2$), $D_6:C_{56}$ ($C_2$), $C_{84}.D_4$ ($C_2$), $C_{84}.Q_8$ ($C_2$)

Order 336: $C_{84}:C_4$ ($C_2^2$) x 2, $C_6:C_{56}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $D_6\times C_{28}$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_2\times C_{168}$ ($C_2^2$)

Order 224: $C_{14}.Q_{16}$ ($S_3$)

Order 168: $D_6\times C_{14}$ ($D_4$), $C_2\times C_{84}$ ($C_2^3$), $C_6:C_{28}$ ($D_4$)

Order 112: $C_{14}:Q_8$ ($D_6$), $C_2\times C_{56}$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $D_6:C_8$ ($D_7$)

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

Order 56: $C_2\times C_{28}$ ($C_2\times D_6$)

Order 48: $C_6:C_8$ ($D_{14}$), $C_4\times D_6$ ($D_{14}$), $C_2\times C_{24}$ ($D_{14}$)

Order 42: $C_{42}$ ($Q_{16}:C_2$), $C_{42}$ ($D_8:C_2$), $C_{42}$ ($C_2^2.D_4$)

Order 28: $C_2\times C_{14}$ ($S_3\times D_4$), $C_{28}$ ($D_4:S_3$), $C_{28}$ ($D_{12}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_6:C_4$ ($D_{28}$), $C_2\times D_6$ ($D_{28}$)

Order 21: $C_{21}$ ($C_2^3.D_4$)

Order 16: $C_2\times C_8$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_6.D_4$), $C_{14}$ ($D_{24}:C_2$), $C_{14}$ ($C_8.D_6$)

Order 12: $C_{12}$ ($D_4:D_7$) x 2, $C_2\times C_6$ ($C_2\times D_{28}$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($(C_2\times C_8).D_6$)

Order 6: $C_6$ ($D_{56}:C_2$), $C_6$ ($C_2^2.D_{28}$), $C_6$ ($C_4.D_{28}$)

Order 4: $C_2^2$ ($S_3\times D_{28}$), $C_4$ ($D_6.D_{14}$), $C_4$ ($D_{12}:D_7$)

Order 3: $C_3$ ($C_2^3.D_{28}$)

Order 2: $C_2$ ($D_6.D_{28}$), $C_2$ ($D_{168}:C_2$), $C_2$ ($C_{24}.D_{14}$)

Order 1: $C_1$ ($(C_2\times C_{24}).D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $(C_2\times C_{24}).D_{14}$ ($C_1$)

Order 672: $C_{84}.D_4$ ($C_2$), $C_{84}.D_4$ ($C_2$), $(S_3\times C_{28}):C_4$ ($C_2$), $C_{168}:C_4$ ($C_2$), $D_6:C_{56}$ ($C_2$), $C_{84}.D_4$ ($C_2$), $C_{84}.Q_8$ ($C_2$)

Order 336: $C_{84}:C_4$ ($C_2^2$) x 2, $C_6:C_{56}$ ($C_2^2$), $C_{28}:C_{12}$ ($C_2^2$), $D_6\times C_{28}$ ($C_2^2$), $C_{42}:Q_8$ ($C_2^2$), $C_2\times C_{168}$ ($C_2^2$)

Order 224: $C_{14}.Q_{16}$ ($S_3$)

Order 168: $D_6\times C_{14}$ ($D_4$), $C_2\times C_{84}$ ($C_2^3$), $C_6:C_{28}$ ($D_4$)

Order 112: $C_{14}:Q_8$ ($D_6$), $C_2\times C_{56}$ ($D_6$), $C_{28}:C_4$ ($D_6$)

Order 96: $D_6:C_8$ ($D_7$)

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

Order 56: $C_2\times C_{28}$ ($C_2\times D_6$)

Order 48: $C_6:C_8$ ($D_{14}$), $C_4\times D_6$ ($D_{14}$), $C_2\times C_{24}$ ($D_{14}$)

Order 42: $C_{42}$ ($Q_{16}:C_2$), $C_{42}$ ($D_8:C_2$), $C_{42}$ ($C_2^2.D_4$)

Order 28: $C_2\times C_{14}$ ($S_3\times D_4$), $C_{28}$ ($D_4:S_3$), $C_{28}$ ($D_{12}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_6:C_4$ ($D_{28}$), $C_2\times D_6$ ($D_{28}$)

Order 21: $C_{21}$ ($C_2^3.D_4$)

Order 16: $C_2\times C_8$ ($S_3\times D_7$)

Order 14: $C_{14}$ ($D_6.D_4$), $C_{14}$ ($D_{24}:C_2$), $C_{14}$ ($C_8.D_6$)

Order 12: $C_{12}$ ($D_4:D_7$) x 2, $C_2\times C_6$ ($C_2\times D_{28}$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($(C_2\times C_8).D_6$)

Order 6: $C_6$ ($D_{56}:C_2$), $C_6$ ($C_2^2.D_{28}$), $C_6$ ($C_4.D_{28}$)

Order 4: $C_2^2$ ($S_3\times D_{28}$), $C_4$ ($D_6.D_{14}$), $C_4$ ($D_{12}:D_7$)

Order 3: $C_3$ ($C_2^3.D_{28}$)

Order 2: $C_2$ ($D_6.D_{28}$), $C_2$ ($D_{168}:C_2$), $C_2$ ($C_{24}.D_{14}$)

Order 1: $C_1$ ($(C_2\times C_{24}).D_{14}$)

Series

Derived series

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

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$(S_3\times C_{28}):C_4$

$\rhd$

$D_6\times C_{28}$

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2\times C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

Character theory

Complex character table

See the $141 \times 141$ 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 $40 \times 40$ rational character table.