Abstract group 1248.1158: $C_2\times C_4\times F_{13}$

• Label: 1248.1158
• Order: \( 2^{5} \cdot 3 \cdot 13 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 13 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $19$
• Trans deg.: $104$
• Rank: $3$

Group information

Description:	$C_2\times C_4\times F_{13}$
Order:	\(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \)
Exponent:	\(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \)
Automorphism group:	$C_2\wr C_2^2\times F_{13}$, of order \(9984\)\(\medspace = 2^{8} \cdot 3 \cdot 13 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{13}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12	13	26	52
Elements	1	55	26	264	182	624	12	36	48	1248
Conjugacy classes	1	7	2	24	14	48	1	3	4	104
Divisions	1	7	1	12	7	12	1	3	2	46
Autjugacy classes	1	4	2	4	8	8	1	2	1	31

Dimension	1	2	4	12	24
Irr. complex chars.	96	0	0	8	0	104
Irr. rational chars.	8	20	12	4	2	46

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$104$
Rank:	$3$
Inequivalent generating triples:	$61152$

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:	$\langle a, b, c \mid a^{4}=b^{12}=c^{26}=[a,b]=[a,c]=1, c^{b}=c^{15} \rangle$
Permutation group:	Degree $19$ $\langle(2,3,5,8)(4,7,10,13)(6,9,12,11)(14,15,17,16), (18,19), (14,16,17,15), (2,4,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 5 & 11 & 11 \\ 2 & 1 & 11 \\ 4 & 9 & 12 \end{array}\right), \left(\begin{array}{rrr} 10 & 4 & 6 \\ 5 & 6 & 8 \\ 4 & 12 & 12 \end{array}\right), \left(\begin{array}{rrr} 4 & 10 & 10 \\ 0 & 1 & 0 \\ 3 & 10 & 11 \end{array}\right), \left(\begin{array}{rrr} 6 & 4 & 4 \\ 9 & 1 & 4 \\ 5 & 8 & 5 \end{array}\right), \left(\begin{array}{rrr} 12 & 0 & 0 \\ 0 & 12 & 0 \\ 0 & 0 & 12 \end{array}\right), \left(\begin{array}{rrr} 0 & 9 & 9 \\ 4 & 5 & 9 \\ 8 & 5 & 1 \end{array}\right), \left(\begin{array}{rrr} 5 & 1 & 12 \\ 4 & 5 & 9 \\ 0 & 10 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{13})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $F_{13}$
Semidirect product:	$(C_{52}:C_6)$ $\,\rtimes\,$ $C_4$	$(C_{52}:C_4)$ $\,\rtimes\,$ $C_6$	$(C_4\times D_{13})$ $\,\rtimes\,$ $C_{12}$	$D_{13}$ $\,\rtimes\,$ $(C_4\times C_{12})$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times D_{26})$ . $C_6$	$(D_{26}:C_4)$ . $C_6$	$(D_{26}:C_{12})$ . $C_2$	$D_{26}$ . $(C_2\times C_{12})$ (2)	all 22
Aut. group:	$\Aut(C_{13}:C_{16})$	$\Aut(C_{13}:C_{20})$	$\Aut(C_5\times D_{26})$	$\Aut(C_{15}\times D_{13})$	all 15

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{12} \simeq C_{2} \times C_{4}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 1416 subgroups in 216 conjugacy classes, 116 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $F_{13}$
Commutator:	$G' \simeq$ $C_{13}$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times F_{13}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{52}$	$G/\operatorname{Fit} \simeq$ $C_{12}$
Radical:	$R \simeq$ $C_2\times C_4\times F_{13}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{26}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1248: $C_2\times C_4\times F_{13}$

Order 624: $C_4\times F_{13}$ x 4, $C_2^2\times F_{13}$ x 2, $D_{26}:C_{12}$

Order 416: $C_{26}:C_4^2$

Order 312: $C_2\times F_{13}$ x 12, $C_{52}:C_6$ x 4, $D_{26}:C_6$, $C_{26}:C_{12}$, $C_{26}:C_{12}$

Order 208: $C_{52}:C_4$ x 4, $D_{26}:C_4$ x 2, $C_4\times D_{26}$

Order 156: $F_{13}$ x 8, $C_{26}:C_6$ x 6, $C_{13}:C_{12}$ x 2, $C_{13}:C_{12}$ x 2, $C_{26}:C_6$

Order 104: $C_{26}:C_4$ x 12, $C_4\times D_{13}$ x 4, $C_2\times C_{52}$, $C_{26}:C_4$, $C_2\times D_{26}$

Order 96: $C_2\times C_4\times C_{12}$

Order 78: $C_{13}:C_6$ x 4, $C_{13}:C_6$ x 3

Order 52: $C_{13}:C_4$ x 8, $D_{26}$ x 6, $C_{52}$ x 2, $C_{13}:C_4$ x 2, $C_2\times C_{26}$

Order 48: $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

Order 39: $C_{13}:C_3$

Order 32: $C_2\times C_4^2$

Order 26: $D_{13}$ x 4, $C_{26}$ x 3

Order 24: $C_2\times C_{12}$ x 18, $C_2^2\times C_6$

Order 16: $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 13: $C_{13}$

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 18, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1248: $C_2\times C_4\times F_{13}$

Order 624: $C_2^2\times F_{13}$, $D_{26}:C_{12}$, $C_4\times F_{13}$

Order 416: $C_{26}:C_4^2$

Order 312: $C_{52}:C_6$ x 2, $C_2\times F_{13}$ x 2, $D_{26}:C_6$, $C_{26}:C_{12}$, $C_{26}:C_{12}$

Order 208: $D_{26}:C_4$, $C_4\times D_{26}$, $C_{52}:C_4$

Order 156: $C_{26}:C_6$ x 3, $F_{13}$, $C_{13}:C_{12}$, $C_{26}:C_6$, $C_{13}:C_{12}$

Order 104: $C_4\times D_{13}$ x 2, $C_{26}:C_4$ x 2, $C_2\times C_{52}$, $C_{26}:C_4$, $C_2\times D_{26}$

Order 96: $C_2\times C_4\times C_{12}$

Order 78: $C_{13}:C_6$ x 2, $C_{13}:C_6$ x 2

Order 52: $D_{26}$ x 3, $C_2\times C_{26}$, $C_{13}:C_4$, $C_{52}$, $C_{13}:C_4$

Order 48: $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$

Order 39: $C_{13}:C_3$

Order 32: $C_2\times C_4^2$

Order 26: $C_{26}$ x 2, $D_{13}$ x 2

Order 24: $C_2\times C_{12}$ x 6, $C_2^2\times C_6$

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

Order 13: $C_{13}$

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 3

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1248: $C_2\times C_4\times F_{13}$ ($C_1$)

Order 624: $C_4\times F_{13}$ ($C_2$) x 4, $C_2^2\times F_{13}$ ($C_2$) x 2, $D_{26}:C_{12}$ ($C_2$)

Order 416: $C_{26}:C_4^2$ ($C_3$)

Order 312: $C_2\times F_{13}$ ($C_4$) x 8, $C_2\times F_{13}$ ($C_2^2$) x 4, $C_{52}:C_6$ ($C_2^2$) x 2, $C_{52}:C_6$ ($C_4$) x 2, $D_{26}:C_6$ ($C_2^2$), $C_{26}:C_{12}$ ($C_4$), $C_{26}:C_{12}$ ($C_4$)

Order 208: $C_{52}:C_4$ ($C_6$) x 4, $D_{26}:C_4$ ($C_6$) x 2, $C_4\times D_{26}$ ($C_6$)

Order 156: $F_{13}$ ($C_2\times C_4$) x 8, $C_{26}:C_6$ ($C_2\times C_4$) x 5, $C_{13}:C_{12}$ ($C_2\times C_4$) x 2, $C_{13}:C_{12}$ ($C_2\times C_4$) x 2, $C_{26}:C_6$ ($C_2^3$), $C_{26}:C_6$ ($C_2\times C_4$)

Order 104: $C_{26}:C_4$ ($C_{12}$) x 8, $C_{26}:C_4$ ($C_2\times C_6$) x 4, $C_4\times D_{13}$ ($C_2\times C_6$) x 2, $C_4\times D_{13}$ ($C_{12}$) x 2, $C_2\times C_{52}$ ($C_{12}$), $C_{26}:C_4$ ($C_{12}$), $C_2\times D_{26}$ ($C_2\times C_6$)

Order 78: $C_{13}:C_6$ ($C_4^2$) x 2, $C_{13}:C_6$ ($C_4^2$) x 2, $C_{13}:C_6$ ($C_2^2\times C_4$) x 2, $C_{13}:C_6$ ($C_2^2\times C_4$)

Order 52: $C_{13}:C_4$ ($C_2\times C_{12}$) x 8, $D_{26}$ ($C_2\times C_{12}$) x 5, $C_{52}$ ($C_2\times C_{12}$) x 2, $C_{13}:C_4$ ($C_2\times C_{12}$) x 2, $C_2\times C_{26}$ ($C_2\times C_{12}$), $D_{26}$ ($C_2^2\times C_6$)

Order 39: $C_{13}:C_3$ ($C_2\times C_4^2$)

Order 26: $C_{26}$ ($C_4\times C_{12}$) x 2, $D_{13}$ ($C_2^2\times C_{12}$) x 2, $D_{13}$ ($C_4\times C_{12}$) x 2, $C_{26}$ ($C_2^2\times C_{12}$)

Order 13: $C_{13}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2\times C_4$ ($F_{13}$)

Order 4: $C_4$ ($C_2\times F_{13}$) x 2, $C_2^2$ ($C_2\times F_{13}$)

Order 2: $C_2$ ($C_4\times F_{13}$) x 2, $C_2$ ($C_2^2\times F_{13}$)

Order 1: $C_1$ ($C_2\times C_4\times F_{13}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1248: $C_2\times C_4\times F_{13}$ ($C_1$)

Order 624: $C_2^2\times F_{13}$ ($C_2$), $D_{26}:C_{12}$ ($C_2$), $C_4\times F_{13}$ ($C_2$)

Order 416: $C_{26}:C_4^2$ ($C_3$)

Order 312: $C_{52}:C_6$ ($C_2^2$), $C_{52}:C_6$ ($C_4$), $D_{26}:C_6$ ($C_2^2$), $C_2\times F_{13}$ ($C_2^2$), $C_2\times F_{13}$ ($C_4$), $C_{26}:C_{12}$ ($C_4$), $C_{26}:C_{12}$ ($C_4$)

Order 208: $D_{26}:C_4$ ($C_6$), $C_4\times D_{26}$ ($C_6$), $C_{52}:C_4$ ($C_6$)

Order 156: $C_{26}:C_6$ ($C_2\times C_4$) x 2, $C_{26}:C_6$ ($C_2^3$), $F_{13}$ ($C_2\times C_4$), $C_{13}:C_{12}$ ($C_2\times C_4$), $C_{26}:C_6$ ($C_2\times C_4$), $C_{13}:C_{12}$ ($C_2\times C_4$)

Order 104: $C_2\times C_{52}$ ($C_{12}$), $C_{26}:C_4$ ($C_{12}$), $C_4\times D_{13}$ ($C_2\times C_6$), $C_4\times D_{13}$ ($C_{12}$), $C_2\times D_{26}$ ($C_2\times C_6$), $C_{26}:C_4$ ($C_2\times C_6$), $C_{26}:C_4$ ($C_{12}$)

Order 78: $C_{13}:C_6$ ($C_4^2$), $C_{13}:C_6$ ($C_2^2\times C_4$), $C_{13}:C_6$ ($C_4^2$), $C_{13}:C_6$ ($C_2^2\times C_4$)

Order 52: $D_{26}$ ($C_2\times C_{12}$) x 2, $C_2\times C_{26}$ ($C_2\times C_{12}$), $D_{26}$ ($C_2^2\times C_6$), $C_{13}:C_4$ ($C_2\times C_{12}$), $C_{52}$ ($C_2\times C_{12}$), $C_{13}:C_4$ ($C_2\times C_{12}$)

Order 39: $C_{13}:C_3$ ($C_2\times C_4^2$)

Order 26: $C_{26}$ ($C_2^2\times C_{12}$), $C_{26}$ ($C_4\times C_{12}$), $D_{13}$ ($C_2^2\times C_{12}$), $D_{13}$ ($C_4\times C_{12}$)

Order 13: $C_{13}$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2\times C_4$ ($F_{13}$)

Order 4: $C_2^2$ ($C_2\times F_{13}$), $C_4$ ($C_2\times F_{13}$)

Order 2: $C_2$ ($C_2^2\times F_{13}$), $C_2$ ($C_4\times F_{13}$)

Order 1: $C_1$ ($C_2\times C_4\times F_{13}$)

Series

Derived series

$C_2\times C_4\times F_{13}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Chief series

$C_2\times C_4\times F_{13}$

$\rhd$

$D_{26}:C_{12}$

$\rhd$

$D_{26}:C_6$

$\rhd$

$C_{26}:C_6$

$\rhd$

$C_{13}:C_6$

$\rhd$

$C_{26}$

$\rhd$

$C_{13}$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4\times F_{13}$

$\rhd$

$C_{13}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $104 \times 104$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.