Abstract group 224.82: $C_7:(C_4\times Q_8)$

• Label: 224.82
• Order: \( 2^{5} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $224$
• Rank: $3$

Group information

Description:	$C_7:(C_4\times Q_8)$
Order:	\(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$F_7\times C_2^4.C_2^3$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_7$
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	14	28
Elements	1	3	124	6	18	72	224
Conjugacy classes	1	3	16	3	9	18	50
Divisions	1	3	11	1	3	4	23
Autjugacy classes	1	3	5	1	3	2	15

Dimension	1	2	4	6	12
Irr. complex chars.	16	28	6	0	0	50
Irr. rational chars.	8	6	1	4	4	23

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$224$
Rank:	$3$
Inequivalent generating triples:	$672$

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:	$\langle a, b, c \mid b^{4}=c^{28}=[a,b]=1, a^{2}=b^{2}c^{14}, c^{a}=c^{13}, c^{b}=c^{15} \rangle$
Permutation group:	Degree $19$ $\langle(9,10,12,11)(14,15)(16,17)(18,19), (1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7)(9,11,12,10), (1,4)(2,6)(3,8)(5,7), (9,12)(10,11), (13,14,16,18,19,17,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 26 \\ 26 & 38 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 22 & 12 \\ 6 & 37 \end{array}\right), \left(\begin{array}{rr} 14 & 0 \\ 0 & 14 \end{array}\right), \left(\begin{array}{rr} 34 & 13 \\ 13 & 8 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/39\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_7:Q_8)$ $\,\rtimes\,$ $C_4$ (4)	$(C_7:C_4)$ $\,\rtimes\,$ $Q_8$ (2)	$C_7$ $\,\rtimes\,$ $(C_4\times Q_8)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(Q_8\times D_7)$	$C_2$ . $(D_4:D_7)$	$(C_4:C_4)$ . $D_7$	$C_4$ . $(C_4\times D_7)$ (2)	all 19

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 214 subgroups in 70 conjugacy classes, 43 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_{14}$
Commutator:	$G' \simeq$ $C_{14}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4:C_{28}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_7:(C_4\times Q_8)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times Q_8$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 224: $C_7:(C_4\times Q_8)$

Order 112: $C_{28}:C_4$ x 3, $C_{14}.D_4$ x 2, $C_{14}:Q_8$, $C_4:C_{28}$

Order 56: $C_{14}:C_4$ x 4, $C_7:Q_8$ x 4, $C_2\times C_{28}$ x 3

Order 32: $C_4\times Q_8$

Order 28: $C_7:C_4$ x 7, $C_{28}$ x 4, $C_2\times C_{14}$

Order 16: $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$

Order 14: $C_{14}$ x 3

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4

Order 7: $C_7$

Order 4: $C_4$ x 11, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 224: $C_7:(C_4\times Q_8)$

Order 112: $C_{28}:C_4$ x 2, $C_{14}:Q_8$, $C_4:C_{28}$, $C_{14}.D_4$

Order 56: $C_{14}:C_4$ x 3, $C_2\times C_{28}$ x 2, $C_7:Q_8$

Order 32: $C_4\times Q_8$

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

Order 16: $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times Q_8$

Order 14: $C_{14}$ x 3

Order 8: $C_2\times C_4$ x 5, $Q_8$

Order 7: $C_7$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 224: $C_7:(C_4\times Q_8)$ ($C_1$)

Order 112: $C_{28}:C_4$ ($C_2$) x 3, $C_{14}.D_4$ ($C_2$) x 2, $C_{14}:Q_8$ ($C_2$), $C_4:C_{28}$ ($C_2$)

Order 56: $C_{14}:C_4$ ($C_2^2$) x 4, $C_7:Q_8$ ($C_4$) x 4, $C_2\times C_{28}$ ($C_2^2$) x 3

Order 28: $C_7:C_4$ ($C_2\times C_4$) x 4, $C_{28}$ ($C_2\times C_4$) x 2, $C_7:C_4$ ($Q_8$) x 2, $C_2\times C_{14}$ ($C_2^3$)

Order 16: $C_4:C_4$ ($D_7$)

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

Order 8: $C_2\times C_4$ ($D_{14}$) x 3

Order 7: $C_7$ ($C_4\times Q_8$)

Order 4: $C_4$ ($C_4\times D_7$) x 2, $C_2^2$ ($C_2\times D_{14}$)

Order 2: $C_2$ ($Q_8\times D_7$), $C_2$ ($D_4:D_7$), $C_2$ ($C_4\times D_{14}$)

Order 1: $C_1$ ($C_7:(C_4\times Q_8)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 224: $C_7:(C_4\times Q_8)$ ($C_1$)

Order 112: $C_{28}:C_4$ ($C_2$) x 2, $C_{14}:Q_8$ ($C_2$), $C_4:C_{28}$ ($C_2$), $C_{14}.D_4$ ($C_2$)

Order 56: $C_{14}:C_4$ ($C_2^2$) x 3, $C_2\times C_{28}$ ($C_2^2$) x 2, $C_7:Q_8$ ($C_4$)

Order 28: $C_2\times C_{14}$ ($C_2^3$), $C_{28}$ ($C_2\times C_4$), $C_7:C_4$ ($Q_8$), $C_7:C_4$ ($C_2\times C_4$)

Order 16: $C_4:C_4$ ($D_7$)

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

Order 8: $C_2\times C_4$ ($D_{14}$) x 2

Order 7: $C_7$ ($C_4\times Q_8$)

Order 4: $C_2^2$ ($C_2\times D_{14}$), $C_4$ ($C_4\times D_7$)

Order 2: $C_2$ ($Q_8\times D_7$), $C_2$ ($D_4:D_7$), $C_2$ ($C_4\times D_{14}$)

Order 1: $C_1$ ($C_7:(C_4\times Q_8)$)

Series

Derived series

$C_7:(C_4\times Q_8)$

$\rhd$

$C_{14}$

$\rhd$

$C_1$

Chief series

$C_7:(C_4\times Q_8)$

$\rhd$

$C_{28}:C_4$

$\rhd$

$C_{14}:C_4$

$\rhd$

$C_2\times C_{14}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_7:(C_4\times Q_8)$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4:C_4$

Supergroups

This group is a maximal subgroup of 81 larger groups in the database.

This group is a maximal quotient of 33 larger groups in the database.

Character theory

Complex character table

See the $50 \times 50$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.