Abstract group 252.29: $C_{42}:C_6$

• Label: 252.29
• Order: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $12$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_{42}:C_6$
Order:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$D_6\times F_7$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	7	44	128	6	42	12	12	252
Conjugacy classes	1	3	5	9	2	6	2	2	30
Divisions	1	3	3	5	1	3	1	1	18
Autjugacy classes	1	2	5	7	1	2	1	1	20

Dimension	1	2	3	4	6	12
Irr. complex chars.	12	6	8	0	4	0	30
Irr. rational chars.	4	6	0	2	4	2	18

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	8	8

Constructions

Presentation:	$\langle a, b \mid a^{6}=b^{42}=1, b^{a}=b^{23} \rangle$
Permutation group:	Degree $12$ $\langle(9,10), (11,12), (2,3,4)(5,7,6), (8,9,10), (1,2,3,5,4,6,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 8 & 7 \\ 7 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/14\Z)$
Transitive group:	42T42				more information
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_7:C_3)$
Semidirect product:	$C_{42}$ $\,\rtimes\,$ $C_6$	$(S_3\times C_7)$ $\,\rtimes\,$ $C_6$ (2)	$C_7$ $\,\rtimes\,$ $(C_6\times S_3)$	$C_6$ $\,\rtimes\,$ $(C_7:C_6)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 192 subgroups in 44 conjugacy classes, 21 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{21}:C_6$
Commutator:	$G' \simeq$ $C_{21}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{42}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{42}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_{42}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
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

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Classes of subgroups up to conjugation

Order 252: $C_{42}:C_6$

Order 126: $C_{21}:C_6$ x 2, $C_{21}:C_6$

Order 84: $C_{14}:C_6$, $S_3\times C_{14}$

Order 63: $C_{21}:C_3$

Order 42: $C_7:C_6$ x 4, $S_3\times C_7$ x 2, $C_{42}$

Order 36: $C_6\times S_3$

Order 28: $C_2\times C_{14}$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 14: $C_{14}$ x 3

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 7: $C_7$

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 252: $C_{42}:C_6$

Order 126: $C_{21}:C_6$, $C_{21}:C_6$

Order 84: $C_{14}:C_6$, $S_3\times C_{14}$

Order 63: $C_{21}:C_3$

Order 42: $C_7:C_6$ x 3, $C_{42}$, $S_3\times C_7$

Order 36: $C_6\times S_3$

Order 28: $C_2\times C_{14}$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times C_6$, $C_3\times S_3$

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 7: $C_7$

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 252: $C_{42}:C_6$ ($C_1$)

Order 126: $C_{21}:C_6$ ($C_2$) x 2, $C_{21}:C_6$ ($C_2$)

Order 84: $S_3\times C_{14}$ ($C_3$)

Order 63: $C_{21}:C_3$ ($C_2^2$)

Order 42: $S_3\times C_7$ ($C_6$) x 2, $C_{42}$ ($C_6$), $C_7:C_6$ ($S_3$)

Order 21: $C_{21}$ ($C_2\times C_6$), $C_7:C_3$ ($D_6$)

Order 14: $C_{14}$ ($C_3\times S_3$)

Order 12: $D_6$ ($C_7:C_3$)

Order 7: $C_7$ ($C_6\times S_3$)

Order 6: $S_3$ ($C_7:C_6$) x 2, $C_6$ ($C_7:C_6$)

Order 3: $C_3$ ($C_{14}:C_6$)

Order 2: $C_2$ ($C_{21}:C_6$)

Order 1: $C_1$ ($C_{42}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 252: $C_{42}:C_6$ ($C_1$)

Order 126: $C_{21}:C_6$ ($C_2$), $C_{21}:C_6$ ($C_2$)

Order 84: $S_3\times C_{14}$ ($C_3$)

Order 63: $C_{21}:C_3$ ($C_2^2$)

Order 42: $C_{42}$ ($C_6$), $S_3\times C_7$ ($C_6$), $C_7:C_6$ ($S_3$)

Order 21: $C_{21}$ ($C_2\times C_6$), $C_7:C_3$ ($D_6$)

Order 14: $C_{14}$ ($C_3\times S_3$)

Order 12: $D_6$ ($C_7:C_3$)

Order 7: $C_7$ ($C_6\times S_3$)

Order 6: $C_6$ ($C_7:C_6$), $S_3$ ($C_7:C_6$)

Order 3: $C_3$ ($C_{14}:C_6$)

Order 2: $C_2$ ($C_{21}:C_6$)

Order 1: $C_1$ ($C_{42}:C_6$)

Series

Derived series

$C_{42}:C_6$

$\rhd$

$C_{21}$

$\rhd$

$C_1$

Chief series

$C_{42}:C_6$

$\rhd$

$C_{21}:C_6$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{42}:C_6$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 44 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.