Abstract group 1008.117: $C_{21}:C_{48}$

• Label: 1008.117
• Order: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$C_{21}:C_{48}$
Order:	\(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{21}.(C_2^3\times C_{12})$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336
Elements	1	1	44	2	44	6	4	88	6	8	12	176	12	12	352	24	24	48	48	96	1008
Conjugacy classes	1	1	8	2	8	2	4	16	2	8	4	32	4	4	64	8	8	16	16	32	240
Divisions	1	1	4	1	4	1	1	4	1	1	1	4	1	1	4	1	1	1	1	1	35
Autjugacy classes	1	1	3	1	3	1	1	3	1	1	1	3	1	1	3	1	1	1	1	1	30

Dimension	1	2	3	4	6	8	12	16	24	48	96
Irr. complex chars.	144	0	96	0	0	0	0	0	0	0	0	240
Irr. rational chars.	2	9	0	5	2	5	3	4	2	2	1	35

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	96
Arbitrary	3	6	16

Constructions

Presentation:	$\langle a, b \mid a^{3}=b^{336}=1, b^{a}=b^{289} \rangle$
Permutation group:	Degree $26$ $\langle(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16), (17,18,19)(21,22,23)(24,26,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{ll}\alpha^{12} & 0 \\ 0 & \alpha^{12} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{30} & 0 \\ 0 & \alpha^{30} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{29} & \alpha^{21} \\ \alpha^{29} & \alpha^{2} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{32} & 0 \\ 0 & \alpha^{32} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{39} & 0 \\ 0 & \alpha^{39} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{32} & \alpha^{32} \\ \alpha^{38} & \alpha^{33} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{24} & 0 \\ 0 & \alpha^{24} \\ \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{49}) = \GL_{2}(\F_{7}[\alpha]/(\alpha^{2} + 6 \alpha + 3))$
Direct product:	$C_{16}$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_7:C_3)$
Semidirect product:	$C_{336}$ $\,\rtimes\,$ $C_3$	$C_{21}$ $\,\rtimes\,$ $C_{48}$	$C_{112}$ $\,\rtimes\,$ $C_3^2$	$C_7$ $\,\rtimes\,$ $(C_3\times C_{48})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{168}$ . $C_6$	$C_{84}$ . $C_{12}$	$C_{42}$ . $C_{24}$	$(C_7:C_{24})$ . $C_6$ (3)	all 18

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

Homology

Abelianization:	$C_{3} \times C_{48} \simeq C_{16} \times C_{3}^{2}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 180 subgroups in 60 conjugacy classes, 40 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{48}$	$G/Z \simeq$ $C_7:C_3$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_3\times C_{48}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_{21}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{336}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_{21}:C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

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 1008: $C_{21}:C_{48}$

Order 504: $C_{21}:C_{24}$

Order 336: $C_7:C_{48}$ x 3, $C_{336}$

Order 252: $C_{21}:C_{12}$

Order 168: $C_7:C_{24}$ x 3, $C_{168}$

Order 144: $C_3\times C_{48}$

Order 126: $C_{21}:C_6$

Order 112: $C_{112}$

Order 84: $C_7:C_{12}$ x 3, $C_{84}$

Order 72: $C_3\times C_{24}$

Order 63: $C_{21}:C_3$

Order 56: $C_{56}$

Order 48: $C_{48}$ x 4

Order 42: $C_7:C_6$ x 3, $C_{42}$

Order 36: $C_3\times C_{12}$

Order 28: $C_{28}$

Order 24: $C_{24}$ x 4

Order 21: $C_7:C_3$ x 3, $C_{21}$

Order 18: $C_3\times C_6$

Order 16: $C_{16}$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1008: $C_{21}:C_{48}$

Order 504: $C_{21}:C_{24}$

Order 336: $C_{336}$, $C_7:C_{48}$

Order 252: $C_{21}:C_{12}$

Order 168: $C_{168}$, $C_7:C_{24}$

Order 144: $C_3\times C_{48}$

Order 126: $C_{21}:C_6$

Order 112: $C_{112}$

Order 84: $C_{84}$, $C_7:C_{12}$

Order 72: $C_3\times C_{24}$

Order 63: $C_{21}:C_3$

Order 56: $C_{56}$

Order 48: $C_{48}$ x 2

Order 42: $C_{42}$, $C_7:C_6$

Order 36: $C_3\times C_{12}$

Order 28: $C_{28}$

Order 24: $C_{24}$ x 2

Order 21: $C_{21}$, $C_7:C_3$

Order 18: $C_3\times C_6$

Order 16: $C_{16}$

Order 14: $C_{14}$

Order 12: $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 2

Order 4: $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1008: $C_{21}:C_{48}$ ($C_1$)

Order 504: $C_{21}:C_{24}$ ($C_2$)

Order 336: $C_7:C_{48}$ ($C_3$) x 3, $C_{336}$ ($C_3$)

Order 252: $C_{21}:C_{12}$ ($C_4$)

Order 168: $C_7:C_{24}$ ($C_6$) x 3, $C_{168}$ ($C_6$)

Order 126: $C_{21}:C_6$ ($C_8$)

Order 112: $C_{112}$ ($C_3^2$)

Order 84: $C_7:C_{12}$ ($C_{12}$) x 3, $C_{84}$ ($C_{12}$)

Order 63: $C_{21}:C_3$ ($C_{16}$)

Order 56: $C_{56}$ ($C_3\times C_6$)

Order 48: $C_{48}$ ($C_7:C_3$)

Order 42: $C_7:C_6$ ($C_{24}$) x 3, $C_{42}$ ($C_{24}$)

Order 28: $C_{28}$ ($C_3\times C_{12}$)

Order 24: $C_{24}$ ($C_7:C_6$)

Order 21: $C_7:C_3$ ($C_{48}$) x 3, $C_{21}$ ($C_{48}$)

Order 16: $C_{16}$ ($C_{21}:C_3$)

Order 14: $C_{14}$ ($C_3\times C_{24}$)

Order 12: $C_{12}$ ($C_7:C_{12}$)

Order 8: $C_8$ ($C_{21}:C_6$)

Order 7: $C_7$ ($C_3\times C_{48}$)

Order 6: $C_6$ ($C_7:C_{24}$)

Order 4: $C_4$ ($C_{21}:C_{12}$)

Order 3: $C_3$ ($C_7:C_{48}$)

Order 2: $C_2$ ($C_{21}:C_{24}$)

Order 1: $C_1$ ($C_{21}:C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1008: $C_{21}:C_{48}$ ($C_1$)

Order 504: $C_{21}:C_{24}$ ($C_2$)

Order 336: $C_{336}$ ($C_3$), $C_7:C_{48}$ ($C_3$)

Order 252: $C_{21}:C_{12}$ ($C_4$)

Order 168: $C_{168}$ ($C_6$), $C_7:C_{24}$ ($C_6$)

Order 126: $C_{21}:C_6$ ($C_8$)

Order 112: $C_{112}$ ($C_3^2$)

Order 84: $C_{84}$ ($C_{12}$), $C_7:C_{12}$ ($C_{12}$)

Order 63: $C_{21}:C_3$ ($C_{16}$)

Order 56: $C_{56}$ ($C_3\times C_6$)

Order 48: $C_{48}$ ($C_7:C_3$)

Order 42: $C_{42}$ ($C_{24}$), $C_7:C_6$ ($C_{24}$)

Order 28: $C_{28}$ ($C_3\times C_{12}$)

Order 24: $C_{24}$ ($C_7:C_6$)

Order 21: $C_{21}$ ($C_{48}$), $C_7:C_3$ ($C_{48}$)

Order 16: $C_{16}$ ($C_{21}:C_3$)

Order 14: $C_{14}$ ($C_3\times C_{24}$)

Order 12: $C_{12}$ ($C_7:C_{12}$)

Order 8: $C_8$ ($C_{21}:C_6$)

Order 7: $C_7$ ($C_3\times C_{48}$)

Order 6: $C_6$ ($C_7:C_{24}$)

Order 4: $C_4$ ($C_{21}:C_{12}$)

Order 3: $C_3$ ($C_7:C_{48}$)

Order 2: $C_2$ ($C_{21}:C_{24}$)

Order 1: $C_1$ ($C_{21}:C_{48}$)

Series

Derived series

$C_{21}:C_{48}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$C_{21}:C_{48}$

$\rhd$

$C_{336}$

$\rhd$

$C_{168}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{21}:C_{48}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_{48}$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $240 \times 240$ 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 $35 \times 35$ rational character table.