Abstract group 229376.a: $\ASigmaL(1,16384)$

• Label: 229376.a
• Order: \( 2^{15} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{21} \cdot 3 \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \cdot 7 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$\ASigmaL(1,16384)$
Order:	\(229376\)\(\medspace = 2^{15} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	Group of order \(308281344\)\(\medspace = 2^{21} \cdot 3 \cdot 7^{2} \)
Composition factors:	$C_2$ x 15, $C_7$
Derived length:	$2$

This group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.

Group statistics

Order	1	2	4	7	14	28
Elements	1	16511	16256	24576	122880	49152	229376
Conjugacy classes	1	1182	19	6	18	6	1232
Divisions	1	1182	19	1	3	1	1207

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\ASigmaL(1,16384)$
Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n \mid a^{14}=b^{4}=c^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(1,4,6,8,10,11,14,16,18,20,22,23,25,28,2,3,5,7,9,12,13,15,17,19,21,24,26,27), (1,2)(7,8)(11,12)(21,22)(25,26)\rangle$
Transitive group:	28T870	28T897			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{14}$ . $C_{14}$ (3)	$C_2^9$ . $(D_4\times F_8)$	$(C_2^{12}:C_7)$ . $D_4$	$C_2^{12}$ . $(C_7\times D_4)$	all 25

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_{2} \times C_{14} \simeq C_{2}^{2} \times C_{7}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 45 normal subgroups (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times C_2^{12}.C_{14}$
Commutator:	$G' \simeq$ $C_2^{13}$	$G/G' \simeq$ $C_2\times C_{14}$
Frattini:	$\Phi \simeq$ $C_2^7$	$G/\Phi \simeq$ $C_2^5:F_8$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{14}.C_2$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $\ASigmaL(1,16384)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_2^5:F_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{14}.C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 2 (order at least 114688) are shown, as well as all normal subgroups of any index.

Order 229376: $\ASigmaL(1,16384)$

Order 114688: $C_2^7.C_2\wr C_7$, $C_2\times C_2^{12}.C_{14}$, $C_2^{12}.C_{28}$

Order 57344: $C_2^{12}.C_{14}$

Order 32768: $C_2^{14}.C_2$

Order 28672: $C_2^6.(D_4\times F_8)$ x 2, $C_2^{12}:C_7$

Order 16384: $C_2^{14}$, $C_2^{13}.C_2$, $C_2^{13}.C_2$

Order 8192: $C_2^{13}$

Order 4096: $C_2^{11}.C_2$ x 2, $C_2^{12}$

Order 2048: $C_2^{10}.C_2$ x 2, $C_2^{10}.C_2$ x 2, $C_2^{11}$ x 2

Order 1024: $C_2^{10}$ x 2

Order 512: $C_2^9$ x 2, $D_4\times C_2^6$

Order 256: $C_2^8$ x 4, $C_2^6\times C_4$

Order 128: $C_2^7$ x 3

Order 64: $C_2^6$ x 3

Order 32: $C_2^5$ x 2

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$ x 2

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 2 (order at least 114688) are shown, as well as all normal subgroups of any index.

Order 229376: $\ASigmaL(1,16384)$

Order 114688: $C_2^7.C_2\wr C_7$, $C_2\times C_2^{12}.C_{14}$, $C_2^{12}.C_{28}$

Order 57344: $C_2^{12}.C_{14}$

Order 32768: $C_2^{14}.C_2$

Order 28672: $C_2^{12}:C_7$, $C_2^6.(D_4\times F_8)$

Order 16384: $C_2^{14}$, $C_2^{13}.C_2$, $C_2^{13}.C_2$

Order 8192: $C_2^{13}$

Order 4096: $C_2^{11}.C_2$, $C_2^{12}$

Order 2048: $C_2^{10}.C_2$, $C_2^{10}.C_2$, $C_2^{11}$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$, $D_4\times C_2^6$

Order 256: $C_2^8$ x 3, $C_2^6\times C_4$

Order 128: $C_2^7$ x 2

Order 64: $C_2^6$ x 2

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 7: $C_7$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 229376: $\ASigmaL(1,16384)$ ($C_1$)

Order 114688: $C_2^7.C_2\wr C_7$ ($C_2$), $C_2\times C_2^{12}.C_{14}$ ($C_2$), $C_2^{12}.C_{28}$ ($C_2$)

Order 57344: $C_2^{12}.C_{14}$ ($C_2^2$)

Order 32768: $C_2^{14}.C_2$ ($C_7$)

Order 28672: $C_2^{12}:C_7$ ($D_4$)

Order 16384: $C_2^{14}$ ($C_{14}$), $C_2^{13}.C_2$ ($C_{14}$), $C_2^{13}.C_2$ ($C_{14}$)

Order 8192: $C_2^{13}$ ($C_2\times C_{14}$)

Order 4096: $C_2^{11}.C_2$ ($F_8$) x 2, $C_2^{12}$ ($C_7\times D_4$)

Order 2048: $C_2^{10}.C_2$ ($C_2\times F_8$) x 2, $C_2^{10}.C_2$ ($C_2\times F_8$) x 2, $C_2^{11}$ ($C_2\times F_8$) x 2

Order 1024: $C_2^{10}$ ($C_2^2\times F_8$) x 2

Order 512: $C_2^9$ ($D_4\times F_8$) x 2, $D_4\times C_2^6$ ($C_2^3:F_8$)

Order 256: $C_2^8$ ($C_2^4:F_8$) x 2, $C_2^8$ ($C_2\wr C_7$) x 2, $C_2^6\times C_4$ ($C_2\wr C_7$)

Order 128: $C_2^7$ ($C_2^5:F_8$) x 2, $C_2^7$ ($C_2^5:F_8$)

Order 64: $C_2^6$ ($C_2^8.C_{14}$) x 2, $C_2^6$ ($C_2^8.C_{14}$)

Order 32: $C_2^5$ ($C_2^9.C_{14}$) x 2

Order 16: $C_2^4$ ($C_2^{10}.C_{14}$) x 2

Order 8: $C_2^3$ ($C_2^6.(D_4\times F_8)$) x 2

Order 4: $C_2^2$ ($C_2^{12}.C_{14}$)

Order 2: $C_2$ ($C_2\times C_2^{12}.C_{14}$)

Order 1: $C_1$ ($\ASigmaL(1,16384)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 229376: $\ASigmaL(1,16384)$ ($C_1$)

Order 114688: $C_2^7.C_2\wr C_7$ ($C_2$), $C_2\times C_2^{12}.C_{14}$ ($C_2$), $C_2^{12}.C_{28}$ ($C_2$)

Order 57344: $C_2^{12}.C_{14}$ ($C_2^2$)

Order 32768: $C_2^{14}.C_2$ ($C_7$)

Order 28672: $C_2^{12}:C_7$ ($D_4$)

Order 16384: $C_2^{14}$ ($C_{14}$), $C_2^{13}.C_2$ ($C_{14}$), $C_2^{13}.C_2$ ($C_{14}$)

Order 8192: $C_2^{13}$ ($C_2\times C_{14}$)

Order 4096: $C_2^{11}.C_2$ ($F_8$), $C_2^{12}$ ($C_7\times D_4$)

Order 2048: $C_2^{10}.C_2$ ($C_2\times F_8$), $C_2^{10}.C_2$ ($C_2\times F_8$), $C_2^{11}$ ($C_2\times F_8$)

Order 1024: $C_2^{10}$ ($C_2^2\times F_8$)

Order 512: $C_2^9$ ($D_4\times F_8$), $D_4\times C_2^6$ ($C_2^3:F_8$)

Order 256: $C_2^8$ ($C_2\wr C_7$) x 2, $C_2^8$ ($C_2^4:F_8$), $C_2^6\times C_4$ ($C_2\wr C_7$)

Order 128: $C_2^7$ ($C_2^5:F_8$), $C_2^7$ ($C_2^5:F_8$)

Order 64: $C_2^6$ ($C_2^8.C_{14}$), $C_2^6$ ($C_2^8.C_{14}$)

Order 32: $C_2^5$ ($C_2^9.C_{14}$)

Order 16: $C_2^4$ ($C_2^{10}.C_{14}$)

Order 8: $C_2^3$ ($C_2^6.(D_4\times F_8)$)

Order 4: $C_2^2$ ($C_2^{12}.C_{14}$)

Order 2: $C_2$ ($C_2\times C_2^{12}.C_{14}$)

Order 1: $C_1$ ($\ASigmaL(1,16384)$)

Series

Derived series

$\ASigmaL(1,16384)$

$\rhd$

$C_2^{13}$

$\rhd$

$C_1$

Chief series

$\ASigmaL(1,16384)$

$\rhd$

$C_2^7.C_2\wr C_7$

$\rhd$

$C_2^{12}.C_{14}$

$\rhd$

$C_2^{13}$

$\rhd$

$C_2^{10}$

$\rhd$

$C_2^7$

$\rhd$

$C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\ASigmaL(1,16384)$

$\rhd$

$C_2^{13}$

$\rhd$

$C_2^{12}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

The $1232 \times 1232$ character table is not available for this group.

Rational character table

The $1207 \times 1207$ rational character table is not available for this group.