Abstract group 336.227: $C_2^2\times D_{42}$

• Label: 336.227
• Order: \( 2^{4} \cdot 3 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \cdot 7 \)
• Perm deg.: $16$
• Trans deg.: $168$
• Rank: $4$

Group information

Description:	$C_2^2\times D_{42}$
Order:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$S_3\times C_2^3.\PSL(2,7)\times F_7$, of order \(338688\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 7^{2} \)
Composition factors:	$C_2$ x 4, $C_3$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	175	2	14	6	42	12	84	336
Conjugacy classes	1	15	1	7	3	21	6	42	96
Divisions	1	15	1	7	1	7	1	7	40
Autjugacy classes	1	2	1	1	1	1	1	1	9

Dimension	1	2	6	12
Irr. complex chars.	16	80	0	0	96
Irr. rational chars.	16	8	8	8	40

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$168$
Rank:	$4$
Inequivalent generating quadruples:	$11115$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{42}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, d^{b}=d^{41} \rangle$
Permutation group:	Degree $16$ $\langle(2,3)(4,5)(6,7)(8,9)(11,12)(13,14)(15,16), (8,9)(13,14)(15,16), (13,15)(14,16), (13,16)(14,15), (10,11,12), (1,2,4,6,7,5,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 5 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $D_{21}$
Semidirect product:	$C_{42}$ $\,\rtimes\,$ $C_2^3$	$C_{21}$ $\,\rtimes\,$ $C_2^4$	$(C_2\times C_{14})$ $\,\rtimes\,$ $D_6$	$C_{14}$ $\,\rtimes\,$ $(C_2\times D_6)$	all 12
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}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1696 subgroups in 268 conjugacy classes, 115 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $D_{21}$
Commutator:	$G' \simeq$ $C_{21}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times D_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^2\times D_{42}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
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

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 336: $C_2^2\times D_{42}$

Order 168: $C_2\times D_{42}$ x 14, $C_2^2\times C_{42}$

Order 112: $C_2^2\times D_{14}$

Order 84: $D_{42}$ x 28, $C_2\times C_{42}$ x 7

Order 56: $C_2\times D_{14}$ x 14, $C_2^2\times C_{14}$

Order 48: $C_2^2\times D_6$

Order 42: $D_{21}$ x 8, $C_{42}$ x 7

Order 28: $D_{14}$ x 28, $C_2\times C_{14}$ x 7

Order 24: $C_2\times D_6$ x 14, $C_2^2\times C_6$

Order 21: $C_{21}$

Order 16: $C_2^4$

Order 14: $D_7$ x 8, $C_{14}$ x 7

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7

Order 8: $C_2^3$ x 15

Order 7: $C_7$

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 35

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 336: $C_2^2\times D_{42}$

Order 168: $C_2^2\times C_{42}$, $C_2\times D_{42}$

Order 112: $C_2^2\times D_{14}$

Order 84: $C_2\times C_{42}$, $D_{42}$

Order 56: $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 48: $C_2^2\times D_6$

Order 42: $C_{42}$, $D_{21}$

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

Order 24: $C_2^2\times C_6$, $C_2\times D_6$

Order 21: $C_{21}$

Order 16: $C_2^4$

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

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

Order 8: $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 336: $C_2^2\times D_{42}$ ($C_1$)

Order 168: $C_2\times D_{42}$ ($C_2$) x 14, $C_2^2\times C_{42}$ ($C_2$)

Order 84: $D_{42}$ ($C_2^2$) x 28, $C_2\times C_{42}$ ($C_2^2$) x 7

Order 56: $C_2^2\times C_{14}$ ($S_3$)

Order 42: $D_{21}$ ($C_2^3$) x 8, $C_{42}$ ($C_2^3$) x 7

Order 28: $C_2\times C_{14}$ ($D_6$) x 7

Order 24: $C_2^2\times C_6$ ($D_7$)

Order 21: $C_{21}$ ($C_2^4$)

Order 14: $C_{14}$ ($C_2\times D_6$) x 7

Order 12: $C_2\times C_6$ ($D_{14}$) x 7

Order 8: $C_2^3$ ($D_{21}$)

Order 7: $C_7$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_2\times D_{14}$) x 7

Order 4: $C_2^2$ ($D_{42}$) x 7

Order 3: $C_3$ ($C_2^2\times D_{14}$)

Order 2: $C_2$ ($C_2\times D_{42}$) x 7

Order 1: $C_1$ ($C_2^2\times D_{42}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 336: $C_2^2\times D_{42}$ ($C_1$)

Order 168: $C_2^2\times C_{42}$ ($C_2$), $C_2\times D_{42}$ ($C_2$)

Order 84: $C_2\times C_{42}$ ($C_2^2$), $D_{42}$ ($C_2^2$)

Order 56: $C_2^2\times C_{14}$ ($S_3$)

Order 42: $C_{42}$ ($C_2^3$), $D_{21}$ ($C_2^3$)

Order 28: $C_2\times C_{14}$ ($D_6$)

Order 24: $C_2^2\times C_6$ ($D_7$)

Order 21: $C_{21}$ ($C_2^4$)

Order 14: $C_{14}$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($D_{14}$)

Order 8: $C_2^3$ ($D_{21}$)

Order 7: $C_7$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_2\times D_{14}$)

Order 4: $C_2^2$ ($D_{42}$)

Order 3: $C_3$ ($C_2^2\times D_{14}$)

Order 2: $C_2$ ($C_2\times D_{42}$)

Order 1: $C_1$ ($C_2^2\times D_{42}$)

Series

Derived series

$C_2^2\times D_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_1$

Chief series

$C_2^2\times D_{42}$

$\rhd$

$C_2\times D_{42}$

$\rhd$

$D_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2^2\times D_{42}$

$\rhd$

$C_{21}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 30 larger groups in the database.

This group is a maximal quotient of 24 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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