Abstract group 1344.11704: $C_2^5\times F_7$

• Label: 1344.11704
• Order: \( 2^{6} \cdot 3 \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{15} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
• Perm deg.: not computed
• Trans deg.: $224$
• Rank: $6$

Group information

Description:	$C_2^5\times F_7$
Order:	\(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^5.\GL(5,2)\times F_7$, of order \(13439139840\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 31 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	7	14
Elements	1	255	14	882	6	186	1344
Conjugacy classes	1	63	2	126	1	31	224
Divisions	1	63	1	63	1	31	160
Autjugacy classes	1	2	2	4	1	1	11

Dimension	1	2	6
Irr. complex chars.	192	0	32	224
Irr. rational chars.	64	64	32	160

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$224$
Rank:	$6$
Inequivalent generating 6-tuples:	$128465064$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{6}=f^{14}=[a,b]= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 14 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 14 \\ 14 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ ${}^5$ $\, \times\, $ $F_7$
Semidirect product:	$D_7$ $\,\rtimes\,$ $(C_2^4\times C_6)$	$(C_7:C_6)$ $\,\rtimes\,$ $C_2^5$	$(C_7:C_3)$ $\,\rtimes\,$ $C_2^6$	$C_7$ $\,\rtimes\,$ $(C_2^5\times C_6)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{28}.D_4)$	$\Aut(D_{14}:C_8)$	$\Aut(C_{42}.D_4)$	$\Aut(D_{14}:C_{12})$	all 6

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{28}\Z)$.

Homology

Abelianization:	$C_{2}^{5} \times C_{6} \simeq C_{2}^{6} \times C_{3}$
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$1$

Subgroups

There are 42956 subgroups in 11300 conjugacy classes, 6024 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^5$	$G/Z \simeq$ $F_7$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_2^5\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^5\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_2^5\times F_7$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6$
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 1344: $C_2^5\times F_7$

Order 672: $C_2^4\times F_7$ x 62, $C_2^3\times C_{14}:C_6$

Order 448: $C_2^4\times D_{14}$

Order 336: $C_2^3\times F_7$ x 620, $C_2^2\times C_{14}:C_6$ x 31

Order 224: $C_2^3\times D_{14}$ x 62, $C_2^4\times C_{14}$

Order 192: $C_2^5\times C_6$

Order 168: $C_2^2\times F_7$ x 1240, $C_2\times C_{14}:C_6$ x 155

Order 112: $C_2^2\times D_{14}$ x 620, $C_2^3\times C_{14}$ x 31

Order 96: $C_2^4\times C_6$ x 63

Order 84: $C_2\times F_7$ x 496, $C_{14}:C_6$ x 155

Order 64: $C_2^6$

Order 56: $C_2\times D_{14}$ x 1240, $C_2^2\times C_{14}$ x 155

Order 48: $C_2^3\times C_6$ x 651

Order 42: $F_7$ x 32, $C_7:C_6$ x 31

Order 32: $C_2^5$ x 63

Order 28: $D_{14}$ x 496, $C_2\times C_{14}$ x 155

Order 24: $C_2^2\times C_6$ x 1395

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 651

Order 14: $D_7$ x 32, $C_{14}$ x 31

Order 12: $C_2\times C_6$ x 651

Order 8: $C_2^3$ x 1395

Order 7: $C_7$

Order 6: $C_6$ x 63

Order 4: $C_2^2$ x 651

Order 3: $C_3$

Order 2: $C_2$ x 63

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1344: $C_2^5\times F_7$

Order 672: $C_2^3\times C_{14}:C_6$, $C_2^4\times F_7$

Order 448: $C_2^4\times D_{14}$

Order 336: $C_2^2\times C_{14}:C_6$, $C_2^3\times F_7$

Order 224: $C_2^4\times C_{14}$, $C_2^3\times D_{14}$

Order 192: $C_2^5\times C_6$

Order 168: $C_2\times C_{14}:C_6$, $C_2^2\times F_7$

Order 112: $C_2^3\times C_{14}$, $C_2^2\times D_{14}$

Order 96: $C_2^4\times C_6$ x 2

Order 84: $C_{14}:C_6$, $C_2\times F_7$

Order 64: $C_2^6$

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

Order 48: $C_2^3\times C_6$ x 2

Order 42: $C_7:C_6$, $F_7$

Order 32: $C_2^5$ x 2

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

Order 24: $C_2^2\times C_6$ x 2

Order 21: $C_7:C_3$

Order 16: $C_2^4$ x 2

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

Order 12: $C_2\times C_6$ x 2

Order 8: $C_2^3$ x 2

Order 7: $C_7$

Order 6: $C_6$ x 2

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 1344: $C_2^5\times F_7$ ($C_1$)

Order 672: $C_2^4\times F_7$ ($C_2$) x 62, $C_2^3\times C_{14}:C_6$ ($C_2$)

Order 448: $C_2^4\times D_{14}$ ($C_3$)

Order 336: $C_2^3\times F_7$ ($C_2^2$) x 620, $C_2^2\times C_{14}:C_6$ ($C_2^2$) x 31

Order 224: $C_2^3\times D_{14}$ ($C_6$) x 62, $C_2^4\times C_{14}$ ($C_6$)

Order 168: $C_2^2\times F_7$ ($C_2^3$) x 1240, $C_2\times C_{14}:C_6$ ($C_2^3$) x 155

Order 112: $C_2^2\times D_{14}$ ($C_2\times C_6$) x 620, $C_2^3\times C_{14}$ ($C_2\times C_6$) x 31

Order 84: $C_2\times F_7$ ($C_2^4$) x 496, $C_{14}:C_6$ ($C_2^4$) x 155

Order 56: $C_2\times D_{14}$ ($C_2^2\times C_6$) x 1240, $C_2^2\times C_{14}$ ($C_2^2\times C_6$) x 155

Order 42: $F_7$ ($C_2^5$) x 32, $C_7:C_6$ ($C_2^5$) x 31

Order 32: $C_2^5$ ($F_7$)

Order 28: $D_{14}$ ($C_2^3\times C_6$) x 496, $C_2\times C_{14}$ ($C_2^3\times C_6$) x 155

Order 21: $C_7:C_3$ ($C_2^6$)

Order 16: $C_2^4$ ($C_2\times F_7$) x 31

Order 14: $D_7$ ($C_2^4\times C_6$) x 32, $C_{14}$ ($C_2^4\times C_6$) x 31

Order 8: $C_2^3$ ($C_2^2\times F_7$) x 155

Order 7: $C_7$ ($C_2^5\times C_6$)

Order 4: $C_2^2$ ($C_2^3\times F_7$) x 155

Order 2: $C_2$ ($C_2^4\times F_7$) x 31

Order 1: $C_1$ ($C_2^5\times F_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1344: $C_2^5\times F_7$ ($C_1$)

Order 672: $C_2^3\times C_{14}:C_6$ ($C_2$), $C_2^4\times F_7$ ($C_2$)

Order 448: $C_2^4\times D_{14}$ ($C_3$)

Order 336: $C_2^2\times C_{14}:C_6$ ($C_2^2$), $C_2^3\times F_7$ ($C_2^2$)

Order 224: $C_2^4\times C_{14}$ ($C_6$), $C_2^3\times D_{14}$ ($C_6$)

Order 168: $C_2\times C_{14}:C_6$ ($C_2^3$), $C_2^2\times F_7$ ($C_2^3$)

Order 112: $C_2^3\times C_{14}$ ($C_2\times C_6$), $C_2^2\times D_{14}$ ($C_2\times C_6$)

Order 84: $C_{14}:C_6$ ($C_2^4$), $C_2\times F_7$ ($C_2^4$)

Order 56: $C_2^2\times C_{14}$ ($C_2^2\times C_6$), $C_2\times D_{14}$ ($C_2^2\times C_6$)

Order 42: $C_7:C_6$ ($C_2^5$), $F_7$ ($C_2^5$)

Order 32: $C_2^5$ ($F_7$)

Order 28: $C_2\times C_{14}$ ($C_2^3\times C_6$), $D_{14}$ ($C_2^3\times C_6$)

Order 21: $C_7:C_3$ ($C_2^6$)

Order 16: $C_2^4$ ($C_2\times F_7$)

Order 14: $C_{14}$ ($C_2^4\times C_6$), $D_7$ ($C_2^4\times C_6$)

Order 8: $C_2^3$ ($C_2^2\times F_7$)

Order 7: $C_7$ ($C_2^5\times C_6$)

Order 4: $C_2^2$ ($C_2^3\times F_7$)

Order 2: $C_2$ ($C_2^4\times F_7$)

Order 1: $C_1$ ($C_2^5\times F_7$)

Series

Derived series

$C_2^5\times F_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$C_2^5\times F_7$

$\rhd$

$C_2^3\times C_{14}:C_6$

$\rhd$

$C_2^2\times C_{14}:C_6$

$\rhd$

$C_2\times C_{14}:C_6$

$\rhd$

$C_{14}:C_6$

$\rhd$

$C_7:C_6$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2^5\times F_7$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_2^5$

Supergroups

This group is a maximal subgroup of 3 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 $224 \times 224$ 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 $160 \times 160$ rational character table (warning: may be slow to load).