Abstract group 20480.e: $C_2^9.(C_2\times F_5)$

• Label: 20480.e
• Order: \( 2^{12} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $20$
• Trans deg.: $20$
• Rank: $2$

Group information

Description:	$C_2^9.(C_2\times F_5)$
Order:	\(20480\)\(\medspace = 2^{12} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_2^7.C_2\wr F_5$, of order \(81920\)\(\medspace = 2^{14} \cdot 5 \)
Composition factors:	$C_2$ x 12, $C_5$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	4	5	8	10
Elements	1	863	7840	1024	7680	3072	20480
Conjugacy classes	1	29	28	1	12	3	74
Divisions	1	29	26	1	6	2	65
Autjugacy classes	1	22	14	1	4	2	44

Dimension	1	2	4	5	8	10	20	40
Irr. complex chars.	8	2	4	8	0	26	20	6	74
Irr. rational chars.	4	4	2	4	1	20	24	6	65

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	10	10
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid b^{10}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,19,13,16,2,20,14,15)(3,6,12,9)(4,5,11,10)(7,17)(8,18), (1,16,9,3,17,11,6,19,14,7)(2,15,10,4,18,12,5,20,13,8)\rangle$
Transitive group:	20T514	20T530	40T11336	40T11339	all 30
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^8:D_5)$ . $D_4$	$(C_2^9.F_5)$ . $C_2$	$(C_2^9.C_2)$ . $F_5$	$C_2^9$ . $(C_2\times F_5)$	all 15

Elements of the group are displayed as permutations of degree 20.

Homology

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

Subgroups

There are 19 normal subgroups (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8.(C_2\times F_5)$
Commutator:	$G' \simeq$ $C_2^8:C_{10}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $C_2\wr F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9.C_2$	$G/\operatorname{Fit} \simeq$ $F_5$
Radical:	$R \simeq$ $C_2^9.(C_2\times F_5)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2\wr F_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2.C_2^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 128 (order at least 160) are shown, as well as all normal subgroups of any index.

Order 20480: $C_2^9.(C_2\times F_5)$

Order 10240: $C_2^9.F_5$ x 2, $C_2^9.D_{10}$

Order 5120: $C_2^8.D_{10}$ x 2, $C_2^8.D_{10}$ x 2, $C_2^8.F_5$ x 2, $C_2^9.C_{10}$

Order 4096: $C_2^8.C_2.C_2^3$

Order 2560: $C_2^8:D_5$ x 3, $C_2^8.C_{10}$, $C_2^8:C_{10}$

Order 2048: $C_2^7.C_2.C_2^3$ x 2, $C_2^9.C_4$ x 2, $C_2^6.C_2.C_2^4$, $C_2^6.C_2.C_2^4$, $C_2^5.D_4^2$

Order 1280: $C_2^6:F_5$, $C_2^8:C_5$

Order 1024: $C_2^2\wr C_4$ x 8, $C_2^2\wr C_2^2$ x 8, $C_2^8.C_2^2$ x 6, $C_2^5.C_2^3.C_2^2$ x 4, $C_2^5.C_2^3.C_2^2$ x 4, $C_2^7.(C_2\times C_4)$ x 4, $C_2^6.C_4^2$ x 4, $C_2^9.C_2$ x 3, $C_2^6.C_2.C_2^3$ x 2, $C_2^6.C_2.C_2^3$ x 2, $C_2^5.C_2.C_2^4$ x 2, $C_2^5.C_2^3.C_2^2$, $C_2^5.C_2^3.C_2^2$, $C_2^8.C_2^2$, $C_2^7.C_2^3$

Order 640: $C_2\wr F_5$ x 4, $C_2^5:D_{10}$

Order 512: $C_2^6.D_4$ x 38, $C_2^7:C_2^2$ x 32, $C_2^4\wr C_2$ x 28, $C_2^7.C_4$ x 22, $C_2^7:C_4$ x 20, $C_2^8.C_2$ x 13, $C_2^7:C_4$ x 13, $C_2^5:\OD_{16}$ x 12, $C_2\times C_2^4.C_2^4$ x 10, $C_2^6:D_4$ x 10, $C_2^7:C_4$ x 8, $C_2\times C_2^3.C_2^4.C_2$ x 8, $C_2^4.C_2^5$ x 6, $C_2\times C_2^2.C_2^5.C_2$ x 4, $C_2^6.(C_2\times C_4)$ x 4, $C_2\times C_2^6.C_2^2$ x 3, $C_2^5.C_2^4$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $(C_2^2\times D_4).C_2^4$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$, $C_2\times C_2^4.C_2^3.C_2$, $(C_2^3\times D_4).C_2^3$, $C_2^5.C_2^4$, $C_2^9$, $C_4^2:C_2^5$

Order 320: $C_2\wr D_5$ x 7, $C_2^4:F_5$ x 6, $C_2^5:C_{10}$

Order 256: $C_2^5:D_4$ x 316, $C_2^6:C_4$ x 149, $C_2^4.C_2^4$ x 144, $C_2^5:D_4$ x 144, $C_2^6:C_4$ x 70, $C_2^5.D_4$ x 52, $C_2^4:\OD_{16}$ x 42, $C_2^6:C_4$ x 40, $C_2^5:D_4$ x 29, $C_2^8$ x 25, $C_2^5.D_4$ x 20, $C_2^5:D_4$ x 18, $C_2^5.D_4$ x 18, $C_2^6.C_4$ x 16, $C_4^2:C_2^4$ x 15, $C_2^4.C_4^2$ x 15, $C_2^5:C_8$ x 14, $C_2^6:C_4$ x 14, $C_2^6:C_4$ x 10, $C_2^4.(C_2\times D_4)$ x 8, $C_2^4.\OD_{16}$ x 8, $C_2^5:C_8$ x 8, $C_4^2:C_2^4$ x 6, $C_4^2:C_2^4$ x 6, $C_2^5:D_4$ x 6, $C_2^5:Q_8$ x 6, $C_2^6:C_4$ x 6, $C_2^5.D_4$ x 6, $C_2^4:\OD_{16}$ x 6, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5.D_4$ x 4, $C_2^6.C_4$ x 4, $C_4^2:C_2^4$ x 3, $C_2^4.C_4^2$ x 3, $(D_4\times C_2^3):C_4$ x 2, $C_2^5.D_4$ x 2, $(C_2^4\times C_4):C_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^4.C_4^2$ x 2, $(C_2^2\times C_8):D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_2^2\times D_4).D_4$, $C_2^4.C_2^4$, $(C_2^2\times \OD_{16}):C_2^2$, $C_2^4.C_2^4$

Order 160: $C_2^4:D_5$ x 9, $C_2\wr C_5$ x 6

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 160) are shown, as well as all normal subgroups of any index.

Order 20480: $C_2^9.(C_2\times F_5)$

Order 10240: $C_2^9.D_{10}$, $C_2^9.F_5$

Order 5120: $C_2^8.D_{10}$ x 2, $C_2^9.C_{10}$, $C_2^8.D_{10}$, $C_2^8.F_5$

Order 4096: $C_2^8.C_2.C_2^3$

Order 2560: $C_2^8:D_5$ x 2, $C_2^8.C_{10}$, $C_2^8:C_{10}$

Order 2048: $C_2^7.C_2.C_2^3$, $C_2^6.C_2.C_2^4$, $C_2^6.C_2.C_2^4$, $C_2^9.C_4$, $C_2^5.D_4^2$

Order 1280: $C_2^6:F_5$, $C_2^8:C_5$

Order 1024: $C_2^8.C_2^2$ x 6, $C_2^9.C_2$ x 3, $C_2^2\wr C_4$ x 3, $C_2^2\wr C_2^2$ x 3, $C_2^5.C_2^3.C_2^2$ x 2, $C_2^5.C_2^3.C_2^2$ x 2, $C_2^5.C_2.C_2^4$ x 2, $C_2^5.C_2^3.C_2^2$, $C_2^6.C_2.C_2^3$, $C_2^5.C_2^3.C_2^2$, $C_2^8.C_2^2$, $C_2^7.C_2^3$, $C_2^6.C_2.C_2^3$, $C_2^7.(C_2\times C_4)$, $C_2^6.C_4^2$

Order 640: $C_2\wr F_5$ x 2, $C_2^5:D_{10}$

Order 512: $C_2^4\wr C_2$ x 15, $C_2^7:C_2^2$ x 13, $C_2^6.D_4$ x 12, $C_2^8.C_2$ x 10, $C_2^7:C_4$ x 10, $C_2^7.C_4$ x 8, $C_2^7:C_4$ x 8, $C_2\times C_2^4.C_2^4$ x 7, $C_2^6:D_4$ x 7, $C_2^4.C_2^5$ x 6, $C_2^5:\OD_{16}$ x 4, $C_2^7:C_4$ x 4, $C_2\times C_2^3.C_2^4.C_2$ x 4, $C_2\times C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^5.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2^6.(C_2\times C_4)$ x 2, $(C_2^2\times D_4).C_2^4$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_2^4$, $C_2\times C_2^4.C_2^3.C_2$, $C_2\times C_2^4.C_2^3.C_2$, $C_2\times C_2^4.C_2^3.C_2$, $C_2\times C_2^4.C_2^3.C_2$, $(C_2^3\times D_4).C_2^3$, $C_2^5.C_2^4$, $C_2^9$, $C_4^2:C_2^5$

Order 320: $C_2\wr D_5$ x 5, $C_2^4:F_5$ x 2, $C_2^5:C_{10}$

Order 256: $C_2^5:D_4$ x 166, $C_2^6:C_4$ x 71, $C_2^6:C_4$ x 46, $C_2^4.C_2^4$ x 38, $C_2^5:D_4$ x 38, $C_2^5.D_4$ x 28, $C_2^5:D_4$ x 22, $C_2^8$ x 19, $C_2^6:C_4$ x 17, $C_2^4:\OD_{16}$ x 15, $C_2^5.D_4$ x 14, $C_4^2:C_2^4$ x 10, $C_2^5:D_4$ x 9, $C_2^5.D_4$ x 9, $C_2^6:C_4$ x 8, $C_2^5:C_8$ x 7, $C_2^6:C_4$ x 7, $C_2^6.C_4$ x 7, $C_4^2:C_2^4$ x 6, $C_4^2:C_2^4$ x 6, $C_2^5.D_4$ x 6, $C_2^4.C_4^2$ x 6, $C_2^4.(C_2\times D_4)$ x 4, $C_4^2:C_2^4$ x 3, $C_2^5:D_4$ x 3, $C_2^5:Q_8$ x 3, $C_2^6:C_4$ x 3, $C_2^4.C_4^2$ x 3, $C_2^4:\OD_{16}$ x 3, $C_2^5.D_4$ x 2, $(D_4\times C_2^3):C_4$ x 2, $C_2^5.D_4$ x 2, $(C_2^4\times C_4):C_4$ x 2, $C_2^4.\OD_{16}$ x 2, $C_2^5:C_8$ x 2, $C_2^5.D_4$ x 2, $C_2^6.C_4$ x 2, $(C_2^2\times C_8):D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $C_2^4.C_4^2$, $C_2^4.C_4^2$, $C_2^4.C_4^2$, $(C_2^2\times D_4).D_4$, $C_2^4.C_2^4$, $(C_2^2\times \OD_{16}):C_2^2$, $C_2^4.C_2^4$

Order 160: $C_2\wr C_5$ x 5, $C_2^4:D_5$ x 5

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20480: $C_2^9.(C_2\times F_5)$ ($C_1$)

Order 10240: $C_2^9.F_5$ ($C_2$) x 2, $C_2^9.D_{10}$ ($C_2$)

Order 5120: $C_2^9.C_{10}$ ($C_4$), $C_2^8.D_{10}$ ($C_2^2$), $C_2^8.D_{10}$ ($C_4$)

Order 2560: $C_2^8:D_5$ ($D_4$) x 2, $C_2^8:C_{10}$ ($C_2\times C_4$)

Order 1280: $C_2^8:C_5$ ($C_2^2:C_4$)

Order 1024: $C_2^9.C_2$ ($F_5$)

Order 512: $C_2^9$ ($C_2\times F_5$)

Order 256: $C_2^8$ ($D_{10}:C_4$)

Order 64: $C_2^6$ ($C_2^4:F_5$)

Order 32: $C_2^5$ ($C_2\wr F_5$)

Order 16: $C_2^4$ ($C_2^6:F_5$)

Order 2: $C_2$ ($C_2^8.(C_2\times F_5)$)

Order 1: $C_1$ ($C_2^9.(C_2\times F_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20480: $C_2^9.(C_2\times F_5)$ ($C_1$)

Order 10240: $C_2^9.D_{10}$ ($C_2$), $C_2^9.F_5$ ($C_2$)

Order 5120: $C_2^9.C_{10}$ ($C_4$), $C_2^8.D_{10}$ ($C_2^2$), $C_2^8.D_{10}$ ($C_4$)

Order 2560: $C_2^8:C_{10}$ ($C_2\times C_4$), $C_2^8:D_5$ ($D_4$)

Order 1280: $C_2^8:C_5$ ($C_2^2:C_4$)

Order 1024: $C_2^9.C_2$ ($F_5$)

Order 512: $C_2^9$ ($C_2\times F_5$)

Order 256: $C_2^8$ ($D_{10}:C_4$)

Order 64: $C_2^6$ ($C_2^4:F_5$)

Order 32: $C_2^5$ ($C_2\wr F_5$)

Order 16: $C_2^4$ ($C_2^6:F_5$)

Order 2: $C_2$ ($C_2^8.(C_2\times F_5)$)

Order 1: $C_1$ ($C_2^9.(C_2\times F_5)$)

Series

Derived series

$C_2^9.(C_2\times F_5)$

$\rhd$

$C_2^8:C_{10}$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^9.(C_2\times F_5)$

$\rhd$

$C_2^9.F_5$

$\rhd$

$C_2^8.D_{10}$

$\rhd$

$C_2^8:C_{10}$

$\rhd$

$C_2^8:C_5$

$\rhd$

$C_2^8$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_2^9.(C_2\times F_5)$

$\rhd$

$C_2^8:C_{10}$

$\rhd$

$C_2^8:C_5$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Complex character table

See the $74 \times 74$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $65 \times 65$ rational character table.