Abstract group 1344.2337: $C_{24}:D_{28}$

• Label: 1344.2337
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $672$
• Rank: $3$

Group information

Description:	$C_{24}:D_{28}$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^7\times S_3\times F_7$, 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	21	24	28	42	56	84	168
Elements	1	115	2	140	62	6	256	64	18	12	128	168	36	192	48	96	1344
Conjugacy classes	1	7	1	12	7	3	20	8	9	3	16	36	9	48	12	24	216
Divisions	1	7	1	7	5	1	5	3	3	1	3	5	3	3	2	2	52
Autjugacy classes	1	5	1	6	4	1	4	3	3	1	2	4	3	2	2	1	43

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	32	136	48	0	0	0	0	0	0	216
Irr. rational chars.	8	10	9	4	4	8	1	5	3	52

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$672$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{28}=c^{24}=[a,c]=1, b^{a}=b^{27}, c^{b}=c^{17} \rangle$
Permutation group:	Degree $22$ $\langle(2,3)(4,5)(6,7)(9,11)(12,13)(14,16)(15,17)(18,19), (8,9,10,11)(21,22), (8,10) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 1 & 56 \\ 56 & 29 \end{array}\right), \left(\begin{array}{rr} 43 & 42 \\ 21 & 1 \end{array}\right), \left(\begin{array}{rr} 71 & 50 \\ 14 & 69 \end{array}\right), \left(\begin{array}{rr} 55 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 22 & 63 \\ 63 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/84\Z)$
Direct product:	$C_8$ $\, \times\, $ $(C_3:D_{28})$
Semidirect product:	$C_{168}$ $\,\rtimes\,$ $D_4$ (2)	$C_{24}$ $\,\rtimes\,$ $D_{28}$ (2)	$(C_8\times D_{14})$ $\,\rtimes\,$ $S_3$	$D_{14}$ $\,\rtimes\,$ $(S_3\times C_8)$ (2)	all 19
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times D_{14})$ . $D_6$	$(C_6:D_{28})$ . $C_4$	$(C_6.D_{28})$ . $C_4$	$(D_{42}:C_4)$ . $C_4$	all 55

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

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1844 subgroups in 268 conjugacy classes, 92 normal (70 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $S_3\times D_7$
Commutator:	$G' \simeq$ $C_{42}$	$G/G' \simeq$ $C_2^2\times C_8$
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:	$R \simeq$ $C_{24}:D_{28}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1344: $C_{24}:D_{28}$

Order 672: $C_8\times D_{42}$, $C_{24}\times D_{14}$, $D_{42}:C_8$, $C_{12}.D_{28}$, $C_{12}:D_{28}$, $C_{24}:C_{28}$, $C_{12}.D_{28}$

Order 448: $C_8\times D_{28}$

Order 336: $C_8\times D_{21}$ x 2, $D_7\times C_{24}$ x 2, $C_{42}:C_8$, $C_{12}:C_{28}$, $C_6:C_{56}$, $C_{14}:C_{24}$, $C_6.D_{28}$, $D_{42}:C_4$, $C_6.D_{28}$, $C_4\times D_{42}$, $C_{12}\times D_{14}$, $C_6:D_{28}$, $C_2\times C_{168}$

Order 224: $C_8\times D_{14}$ x 2, $D_{14}:C_8$ x 2, $C_4\times D_{28}$, $C_4\times C_{56}$, $C_{28}:C_8$

Order 192: $C_{24}:D_4$

Order 168: $C_3:D_{28}$ x 4, $C_{168}$ x 2, $C_4\times D_{21}$ x 2, $C_6:C_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_2\times D_{42}$, $C_6\times D_{14}$, $C_{21}:C_8$, $C_7:C_{24}$, $C_2\times C_{84}$, $C_{42}:C_4$, $C_3:C_{56}$, $C_{14}:C_{12}$

Order 112: $C_8\times D_7$ x 4, $C_{14}:C_8$ x 2, $C_4\times D_{14}$ x 2, $C_2\times C_{56}$ x 2, $D_{14}:C_4$ x 2, $C_2\times D_{28}$, $C_4\times C_{28}$, $C_{28}:C_4$

Order 96: $C_{12}.D_4$, $D_6:C_8$, $C_{12}.D_4$, $C_{24}:C_4$, $C_2^2\times C_{24}$, $C_{12}:D_4$, $C_8\times D_6$

Order 84: $D_{42}$ x 4, $C_3\times D_{14}$ x 4, $C_3:C_{28}$ x 3, $C_{84}$ x 2, $C_{21}:C_4$, $C_7:C_{12}$, $C_2\times C_{42}$

Order 64: $C_8\times D_4$

Order 56: $D_{28}$ x 4, $C_4\times D_7$ x 4, $C_2\times C_{28}$ x 3, $C_{56}$ x 3, $C_{14}:C_4$ x 2, $C_2\times D_{14}$ x 2, $C_7:C_8$ x 2

Order 48: $C_2\times C_{24}$ x 4, $C_6:C_8$ x 2, $S_3\times C_8$ x 2, $C_2^2\times C_{12}$, $C_6:D_4$, $C_4\times D_6$, $C_6.D_4$, $D_6:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3, $D_{21}$ x 2, $C_3\times D_7$ x 2

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 28: $D_{14}$ x 8, $C_{28}$ x 5, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 4, $C_3:D_4$ x 4, $C_6:C_4$ x 3, $C_{24}$ x 3, $C_4\times S_3$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$, $C_2\times D_6$

Order 21: $C_{21}$

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

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

Order 12: $C_2\times C_6$ x 5, $D_6$ x 4, $C_3:C_4$ x 4, $C_{12}$ x 3

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_{24}:D_{28}$

Order 672: $C_8\times D_{42}$, $C_{24}\times D_{14}$, $D_{42}:C_8$, $C_{12}.D_{28}$, $C_{12}:D_{28}$, $C_{24}:C_{28}$, $C_{12}.D_{28}$

Order 448: $C_8\times D_{28}$

Order 336: $C_{42}:C_8$, $C_8\times D_{21}$, $C_{12}:C_{28}$, $C_6:C_{56}$, $C_{14}:C_{24}$, $D_7\times C_{24}$, $C_6.D_{28}$, $D_{42}:C_4$, $C_6.D_{28}$, $C_4\times D_{42}$, $C_{12}\times D_{14}$, $C_6:D_{28}$, $C_2\times C_{168}$

Order 224: $C_8\times D_{14}$ x 2, $D_{14}:C_8$ x 2, $C_4\times D_{28}$, $C_4\times C_{56}$, $C_{28}:C_8$

Order 192: $C_{24}:D_4$

Order 168: $C_4\times D_{21}$ x 2, $C_6:C_{28}$ x 2, $C_{12}\times D_7$ x 2, $C_{168}$, $C_2\times D_{42}$, $C_6\times D_{14}$, $C_{21}:C_8$, $C_7:C_{24}$, $C_2\times C_{84}$, $C_{42}:C_4$, $C_3:C_{56}$, $C_{14}:C_{12}$, $C_3:D_{28}$

Order 112: $C_{14}:C_8$ x 2, $C_8\times D_7$ x 2, $C_4\times D_{14}$ x 2, $C_2\times C_{56}$ x 2, $D_{14}:C_4$ x 2, $C_2\times D_{28}$, $C_4\times C_{28}$, $C_{28}:C_4$

Order 96: $C_{12}.D_4$, $D_6:C_8$, $C_{12}.D_4$, $C_{24}:C_4$, $C_2^2\times C_{24}$, $C_{12}:D_4$, $C_8\times D_6$

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

Order 64: $C_8\times D_4$

Order 56: $C_4\times D_7$ x 4, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 2, $C_{56}$ x 2, $C_2\times D_{14}$ x 2, $C_7:C_8$ x 2, $D_{28}$

Order 48: $C_2\times C_{24}$ x 3, $C_6:C_8$ x 2, $C_2^2\times C_{12}$, $C_6:D_4$, $S_3\times C_8$, $C_4\times D_6$, $C_6.D_4$, $D_6:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 42: $C_{42}$ x 3, $D_{21}$, $C_3\times D_7$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 28: $D_{14}$ x 6, $C_{28}$ x 4, $C_7:C_4$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$ x 4, $C_6:C_4$ x 3, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $C_2^2\times C_6$, $C_2\times D_6$

Order 21: $C_{21}$

Order 16: $C_2\times C_8$ x 6, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

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

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

Order 8: $C_2\times C_4$ x 9, $C_8$ x 4, $C_2^3$ x 2, $D_4$

Order 7: $C_7$

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

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1344: $C_{24}:D_{28}$ ($C_1$)

Order 672: $C_8\times D_{42}$ ($C_2$), $C_{24}\times D_{14}$ ($C_2$), $D_{42}:C_8$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}:D_{28}$ ($C_2$), $C_{24}:C_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$)

Order 336: $C_{42}:C_8$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_6:C_{56}$ ($C_2^2$), $C_{14}:C_{24}$ ($C_2^2$), $C_6.D_{28}$ ($C_4$), $D_{42}:C_4$ ($C_4$), $C_6.D_{28}$ ($C_4$), $C_4\times D_{42}$ ($C_2^2$), $C_{12}\times D_{14}$ ($C_2^2$), $C_6:D_{28}$ ($C_4$), $C_2\times C_{168}$ ($C_2^2$)

Order 224: $C_8\times D_{14}$ ($S_3$)

Order 168: $C_3:D_{28}$ ($C_8$) x 4, $C_{168}$ ($D_4$) x 2, $C_6:C_{28}$ ($C_2\times C_4$) x 2, $C_2\times D_{42}$ ($C_2\times C_4$), $C_6\times D_{14}$ ($C_2\times C_4$), $C_2\times C_{84}$ ($C_2^3$), $C_{42}:C_4$ ($C_2\times C_4$), $C_{14}:C_{12}$ ($C_2\times C_4$)

Order 112: $C_{14}:C_8$ ($D_6$), $C_4\times D_{14}$ ($D_6$), $C_2\times C_{56}$ ($D_6$)

Order 96: $C_{24}:C_4$ ($D_7$)

Order 84: $C_3:C_{28}$ ($C_2\times C_8$) x 2, $D_{42}$ ($C_2\times C_8$) x 2, $C_3\times D_{14}$ ($C_2\times C_8$) x 2, $C_{84}$ ($D_4:C_2$), $C_{84}$ ($C_2\times D_4$), $C_2\times C_{42}$ ($C_2^2\times C_4$)

Order 56: $C_{56}$ ($C_3:D_4$) x 2, $C_2\times C_{28}$ ($C_2\times D_6$), $C_{14}:C_4$ ($C_4\times S_3$), $C_2\times D_{14}$ ($C_4\times S_3$)

Order 48: $C_6:C_8$ ($D_{14}$), $C_2\times C_{24}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($\OD_{16}:C_2$), $C_{42}$ ($C_2^2\times C_8$), $C_{42}$ ($C_4\times D_4$)

Order 28: $D_{14}$ ($S_3\times C_8$) x 2, $C_2\times C_{14}$ ($C_4\times D_6$), $C_{28}$ ($C_6:D_4$), $C_{28}$ ($D_{12}:C_2$)

Order 24: $C_6:C_4$ ($C_4\times D_7$) x 2, $C_{24}$ ($D_{28}$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$)

Order 21: $C_{21}$ ($C_8\times D_4$)

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

Order 14: $C_{14}$ ($C_{12}:D_4$), $C_{14}$ ($C_8.D_6$), $C_{14}$ ($C_8\times D_6$)

Order 12: $C_3:C_4$ ($C_8\times D_7$) x 2, $C_2\times C_6$ ($C_4\times D_{14}$), $C_{12}$ ($D_{28}:C_2$), $C_{12}$ ($C_2\times D_{28}$)

Order 8: $C_8$ ($C_3:D_{28}$) x 2, $C_2\times C_4$ ($S_3\times D_{14}$)

Order 7: $C_7$ ($C_{24}:D_4$)

Order 6: $C_6$ ($C_8.D_{14}$), $C_6$ ($C_8\times D_{14}$), $C_6$ ($C_4\times D_{28}$)

Order 4: $C_2^2$ ($C_{12}:D_{14}$), $C_4$ ($C_6:D_{28}$), $C_4$ ($C_{12}.D_{14}$)

Order 3: $C_3$ ($C_8\times D_{28}$)

Order 2: $C_2$ ($C_{12}:D_{28}$), $C_2$ ($C_{24}.D_{14}$), $C_2$ ($C_{24}:D_{14}$)

Order 1: $C_1$ ($C_{24}:D_{28}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_{24}:D_{28}$ ($C_1$)

Order 672: $C_8\times D_{42}$ ($C_2$), $C_{24}\times D_{14}$ ($C_2$), $D_{42}:C_8$ ($C_2$), $C_{12}.D_{28}$ ($C_2$), $C_{12}:D_{28}$ ($C_2$), $C_{24}:C_{28}$ ($C_2$), $C_{12}.D_{28}$ ($C_2$)

Order 336: $C_{42}:C_8$ ($C_2^2$), $C_{12}:C_{28}$ ($C_2^2$), $C_6:C_{56}$ ($C_2^2$), $C_{14}:C_{24}$ ($C_2^2$), $C_6.D_{28}$ ($C_4$), $D_{42}:C_4$ ($C_4$), $C_6.D_{28}$ ($C_4$), $C_4\times D_{42}$ ($C_2^2$), $C_{12}\times D_{14}$ ($C_2^2$), $C_6:D_{28}$ ($C_4$), $C_2\times C_{168}$ ($C_2^2$)

Order 224: $C_8\times D_{14}$ ($S_3$)

Order 168: $C_6:C_{28}$ ($C_2\times C_4$) x 2, $C_{168}$ ($D_4$), $C_2\times D_{42}$ ($C_2\times C_4$), $C_6\times D_{14}$ ($C_2\times C_4$), $C_2\times C_{84}$ ($C_2^3$), $C_{42}:C_4$ ($C_2\times C_4$), $C_{14}:C_{12}$ ($C_2\times C_4$), $C_3:D_{28}$ ($C_8$)

Order 112: $C_{14}:C_8$ ($D_6$), $C_4\times D_{14}$ ($D_6$), $C_2\times C_{56}$ ($D_6$)

Order 96: $C_{24}:C_4$ ($D_7$)

Order 84: $C_{84}$ ($D_4:C_2$), $C_{84}$ ($C_2\times D_4$), $C_3:C_{28}$ ($C_2\times C_8$), $C_2\times C_{42}$ ($C_2^2\times C_4$), $D_{42}$ ($C_2\times C_8$), $C_3\times D_{14}$ ($C_2\times C_8$)

Order 56: $C_2\times C_{28}$ ($C_2\times D_6$), $C_{14}:C_4$ ($C_4\times S_3$), $C_{56}$ ($C_3:D_4$), $C_2\times D_{14}$ ($C_4\times S_3$)

Order 48: $C_6:C_8$ ($D_{14}$), $C_2\times C_{24}$ ($D_{14}$), $C_{12}:C_4$ ($D_{14}$)

Order 42: $C_{42}$ ($\OD_{16}:C_2$), $C_{42}$ ($C_2^2\times C_8$), $C_{42}$ ($C_4\times D_4$)

Order 28: $C_2\times C_{14}$ ($C_4\times D_6$), $D_{14}$ ($S_3\times C_8$), $C_{28}$ ($C_6:D_4$), $C_{28}$ ($D_{12}:C_2$)

Order 24: $C_6:C_4$ ($C_4\times D_7$) x 2, $C_2\times C_{12}$ ($C_2\times D_{14}$), $C_{24}$ ($D_{28}$)

Order 21: $C_{21}$ ($C_8\times D_4$)

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

Order 14: $C_{14}$ ($C_{12}:D_4$), $C_{14}$ ($C_8.D_6$), $C_{14}$ ($C_8\times D_6$)

Order 12: $C_2\times C_6$ ($C_4\times D_{14}$), $C_{12}$ ($D_{28}:C_2$), $C_{12}$ ($C_2\times D_{28}$), $C_3:C_4$ ($C_8\times D_7$)

Order 8: $C_2\times C_4$ ($S_3\times D_{14}$), $C_8$ ($C_3:D_{28}$)

Order 7: $C_7$ ($C_{24}:D_4$)

Order 6: $C_6$ ($C_8.D_{14}$), $C_6$ ($C_8\times D_{14}$), $C_6$ ($C_4\times D_{28}$)

Order 4: $C_2^2$ ($C_{12}:D_{14}$), $C_4$ ($C_6:D_{28}$), $C_4$ ($C_{12}.D_{14}$)

Order 3: $C_3$ ($C_8\times D_{28}$)

Order 2: $C_2$ ($C_{12}:D_{28}$), $C_2$ ($C_{24}.D_{14}$), $C_2$ ($C_{24}:D_{14}$)

Order 1: $C_1$ ($C_{24}:D_{28}$)

Series

Derived series

$C_{24}:D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_1$

Chief series

$C_{24}:D_{28}$

$\rhd$

$C_{24}\times D_{14}$

$\rhd$

$C_2\times C_{168}$

$\rhd$

$C_{168}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}:D_{28}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

Character theory

Complex character table

See the $216 \times 216$ 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 $52 \times 52$ rational character table.