Abstract group 448.652: $(C_2\times C_8).D_{14}$

• Label: 448.652
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$(C_2\times C_8).D_{14}$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(56\)\(\medspace = 2^{3} \cdot 7 \)
Automorphism group:	$C_7.(C_2^4\times C_6).C_2^5$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	8	14	28	56
Elements	1	7	120	6	128	42	48	96	448
Conjugacy classes	1	5	14	3	8	15	18	24	88
Divisions	1	5	10	1	4	5	4	2	32
Autjugacy classes	1	3	5	1	2	3	3	1	19

Dimension	1	2	4	6	12	24
Irr. complex chars.	16	60	12	0	0	0	88
Irr. rational chars.	8	8	4	4	6	2	32

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$1344$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	8	12

Constructions

Presentation:	$\langle a, b, c \mid b^{4}=c^{28}=1, a^{4}=c^{14}, b^{a}=b^{3}c^{14}, c^{a}=c^{15}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $19$ $\langle(2,3)(4,5)(6,7)(8,9,10,11)(12,13,15,17)(14,18,19,16), (8,10)(9,11)(14,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 15 & 94 \\ 77 & 90 \end{array}\right), \left(\begin{array}{rr} 85 & 14 \\ 77 & 50 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{28}.D_4)$ $\,\rtimes\,$ $C_2$ (4)	$(C_7:C_4)$ $\,\rtimes\,$ $\OD_{16}$ (4)	$C_7$ $\,\rtimes\,$ $(C_4:\OD_{16})$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_{28})$ . $Q_8$ (2)	$C_{28}$ . $(C_2\times Q_8)$	$C_4$ . $(C_{14}:Q_8)$	$(C_2\times C_{28})$ . $D_4$ (2)	all 37

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 420 subgroups in 126 conjugacy classes, 67 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_2\times C_{14}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{14}\times \OD_{16}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_8).D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:\OD_{16}$
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 448: $(C_2\times C_8).D_{14}$

Order 224: $C_{28}.D_4$ x 4, $C_{14}\times \OD_{16}$, $C_{14}:C_4^2$, $C_{14}:\OD_{16}$

Order 112: $C_{28}:C_4$ x 4, $C_7:\OD_{16}$ x 2, $C_{14}:C_8$ x 2, $C_2^2.D_{14}$ x 2, $C_7\times \OD_{16}$ x 2, $C_2\times C_{56}$ x 2, $C_2^2\times C_{28}$

Order 64: $C_4:\OD_{16}$

Order 56: $C_{14}:C_4$ x 8, $C_2\times C_{28}$ x 6, $C_{56}$ x 2, $C_7:C_8$ x 2, $C_2^2\times C_{14}$

Order 32: $C_4:C_8$ x 4, $C_2\times \OD_{16}$ x 2, $C_2\times C_4^2$

Order 28: $C_7:C_4$ x 6, $C_2\times C_{14}$ x 5, $C_{28}$ x 4

Order 16: $\OD_{16}$ x 4, $C_2\times C_8$ x 4, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 14: $C_{14}$ x 5

Order 8: $C_2\times C_4$ x 14, $C_8$ x 4, $C_2^3$

Order 7: $C_7$

Order 4: $C_4$ x 10, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $(C_2\times C_8).D_{14}$

Order 224: $C_{28}.D_4$, $C_{14}\times \OD_{16}$, $C_{14}:C_4^2$, $C_{14}:\OD_{16}$

Order 112: $C_{28}:C_4$ x 3, $C_7:\OD_{16}$, $C_{14}:C_8$, $C_2^2\times C_{28}$, $C_2^2.D_{14}$, $C_7\times \OD_{16}$, $C_2\times C_{56}$

Order 64: $C_4:\OD_{16}$

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

Order 32: $C_2\times \OD_{16}$ x 2, $C_2\times C_4^2$, $C_4:C_8$

Order 28: $C_2\times C_{14}$ x 3, $C_{28}$ x 3, $C_7:C_4$ x 2

Order 16: $C_4^2$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_2^2\times C_4$ x 2

Order 14: $C_{14}$ x 3

Order 8: $C_2\times C_4$ x 7, $C_8$ x 2, $C_2^3$

Order 7: $C_7$

Order 4: $C_4$ x 5, $C_2^2$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $(C_2\times C_8).D_{14}$ ($C_1$)

Order 224: $C_{28}.D_4$ ($C_2$) x 4, $C_{14}\times \OD_{16}$ ($C_2$), $C_{14}:C_4^2$ ($C_2$), $C_{14}:\OD_{16}$ ($C_2$)

Order 112: $C_{14}:C_8$ ($C_2^2$) x 2, $C_2^2.D_{14}$ ($C_4$) x 2, $C_2\times C_{56}$ ($C_2^2$) x 2, $C_{28}:C_4$ ($C_2^2$) x 2, $C_{28}:C_4$ ($C_4$) x 2, $C_2^2\times C_{28}$ ($C_2^2$)

Order 56: $C_{14}:C_4$ ($C_2\times C_4$) x 4, $C_2\times C_{28}$ ($Q_8$) x 2, $C_2\times C_{28}$ ($D_4$) x 2, $C_2\times C_{28}$ ($C_2^3$), $C_2\times C_{28}$ ($C_2\times C_4$), $C_2^2\times C_{14}$ ($C_2\times C_4$)

Order 32: $C_2\times \OD_{16}$ ($D_7$)

Order 28: $C_7:C_4$ ($\OD_{16}$) x 4, $C_2\times C_{14}$ ($C_4:C_4$) x 2, $C_{28}$ ($C_4:C_4$) x 2, $C_2\times C_{14}$ ($C_2^2\times C_4$), $C_{28}$ ($C_2\times Q_8$), $C_{28}$ ($C_2\times D_4$)

Order 16: $C_2\times C_8$ ($D_{14}$) x 2, $C_2^2\times C_4$ ($D_{14}$)

Order 14: $C_{14}$ ($C_2\times \OD_{16}$) x 2, $C_{14}$ ($C_2^2.D_4$)

Order 8: $C_2\times C_4$ ($C_7:D_4$) x 2, $C_2\times C_4$ ($C_7:Q_8$) x 2, $C_2^3$ ($C_4\times D_7$), $C_2\times C_4$ ($C_4\times D_7$), $C_2\times C_4$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_4:\OD_{16}$)

Order 4: $C_2^2$ ($C_{14}.D_4$) x 2, $C_4$ ($C_{14}.D_4$) x 2, $C_2^2$ ($C_4\times D_{14}$), $C_4$ ($C_{14}:D_4$), $C_4$ ($C_{14}:Q_8$)

Order 2: $C_2$ ($D_7\times \OD_{16}$) x 2, $C_2$ ($C_2^3.D_{14}$)

Order 1: $C_1$ ($(C_2\times C_8).D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $(C_2\times C_8).D_{14}$ ($C_1$)

Order 224: $C_{28}.D_4$ ($C_2$), $C_{14}\times \OD_{16}$ ($C_2$), $C_{14}:C_4^2$ ($C_2$), $C_{14}:\OD_{16}$ ($C_2$)

Order 112: $C_{28}:C_4$ ($C_4$) x 2, $C_{14}:C_8$ ($C_2^2$), $C_2^2\times C_{28}$ ($C_2^2$), $C_2^2.D_{14}$ ($C_4$), $C_2\times C_{56}$ ($C_2^2$), $C_{28}:C_4$ ($C_2^2$)

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

Order 32: $C_2\times \OD_{16}$ ($D_7$)

Order 28: $C_2\times C_{14}$ ($C_4:C_4$), $C_2\times C_{14}$ ($C_2^2\times C_4$), $C_{28}$ ($C_4:C_4$), $C_{28}$ ($C_2\times Q_8$), $C_{28}$ ($C_2\times D_4$), $C_7:C_4$ ($\OD_{16}$)

Order 16: $C_2\times C_8$ ($D_{14}$), $C_2^2\times C_4$ ($D_{14}$)

Order 14: $C_{14}$ ($C_2\times \OD_{16}$), $C_{14}$ ($C_2^2.D_4$)

Order 8: $C_2^3$ ($C_4\times D_7$), $C_2\times C_4$ ($C_7:D_4$), $C_2\times C_4$ ($C_4\times D_7$), $C_2\times C_4$ ($C_7:Q_8$), $C_2\times C_4$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_4:\OD_{16}$)

Order 4: $C_2^2$ ($C_4\times D_{14}$), $C_2^2$ ($C_{14}.D_4$), $C_4$ ($C_{14}:D_4$), $C_4$ ($C_{14}:Q_8$), $C_4$ ($C_{14}.D_4$)

Order 2: $C_2$ ($C_2^3.D_{14}$), $C_2$ ($D_7\times \OD_{16}$)

Order 1: $C_1$ ($(C_2\times C_8).D_{14}$)

Series

Derived series

$(C_2\times C_8).D_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_1$

Chief series

$(C_2\times C_8).D_{14}$

$\rhd$

$C_{28}.D_4$

$\rhd$

$C_2\times C_{56}$

$\rhd$

$C_2\times C_{28}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_8).D_{14}$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times \OD_{16}$

Supergroups

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

This group is a maximal quotient of 5 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.