Abstract group 20480.rp: $C_4\wr C_5.C_4$

• Label: 20480.rp
• Order: \( 2^{12} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_4\wr C_5.C_4$
Order:	\(20480\)\(\medspace = 2^{12} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_3\times \SL(2,13)$, of order \(983040\)\(\medspace = 2^{16} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 12, $C_5$
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	5	8	10	20	40
Elements	1	63	1984	1024	2048	3072	4096	8192	20480
Conjugacy classes	1	11	204	4	16	8	12	16	272
Divisions	1	11	104	1	8	2	2	2	131
Autjugacy classes	1	5	9	4	2	8	8	4	41

Dimension	1	2	4	5	8	10	16	20
Irr. complex chars.	40	10	0	24	0	198	0	0	272
Irr. rational chars.	4	2	5	12	2	6	1	99	131

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	20	20
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{4}=c^{4}=d^{4}=e^{4}=f^{2}=g^{2}=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,23,13,36,28,3,22,16,34,26,2,24,14,35,27,4,21,15,33,25)(5,20,10,37,30,8,18,11,40,32,6,19,9,38,29,7,17,12,39,31) \!\cdots\! \rangle$
Transitive group:	40T12795				more information
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:	$(C_4\wr C_5)$ . $C_4$	$(C_4^4:C_{40})$ . $C_2$	$(C_2\times C_4^5)$ . $C_{10}$ (2)	$C_4^5$ . $(C_2\times C_{10})$	all 28

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

Homology

Abelianization:	$C_{2} \times C_{20} \simeq C_{2} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 604086 subgroups in 59301 conjugacy classes, 33 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_2\times C_4^4:C_{10}$
Commutator:	$G' \simeq$ $C_2\times C_4^4$	$G/G' \simeq$ $C_2\times C_{20}$
Frattini:	$\Phi \simeq$ $C_2^4\times C_4$	$G/\Phi \simeq$ $C_2^5:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^5.C_2^2$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_4\wr C_5.C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^5:C_{20}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^5.C_2^2$
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: $C_4\wr C_5.C_4$

Order 10240: $C_4^4:C_{40}$ x 2, $C_2\times C_4^4.C_{20}$

Order 5120: $C_4\wr C_5$ x 2, $C_2\times C_4^4:C_{10}$

Order 4096: $C_4^5.C_2^2$

Order 2560: $C_4^4:C_{10}$ x 2

Order 2048: $C_4^5.C_2$ x 8, $(C_2^2\times C_4^4).C_2$ x 6, $C_2\times C_4^5$

Order 1280: $C_2^6.C_{20}$, $C_4^4:C_5$

Order 1024: $(C_2\times C_4^4).C_2$ x 96, $(C_2^3\times C_4^3).C_2$ x 16, $C_2\times (C_2^2\times C_4^3).C_2$ x 12, $C_4^5$ x 8, $C_2^2\times C_4^4$ x 7

Order 640: $C_2^4:C_{40}$ x 2, $C_2^5:C_{20}$

Order 512: $C_2\times C_4^4$ x 200, $(C_2^2\times C_4^3).C_2$ x 128, $(C_2^2\times C_4^3).C_2$ x 96, $C_2\times (C_2\times C_4^3).C_2$ x 96, $C_2^3\times C_4^3$ x 31, $C_2^2\times (C_2^2\times C_4^2).C_2$ x 24

Order 320: $C_2^4:C_{20}$ x 2, $C_2^5:C_{10}$

Order 256: $C_2^2\times C_4^3$ x 1016, $C_4^4$ x 820, $(C_2\times C_4^2):C_8$ x 384, $C_2^4.C_4^2$ x 288, $C_2^4.C_4^2$ x 96, $C_2^4\times C_4^2$ x 31, $C_2^4\times \OD_{16}$ x 4

Order 160: $C_2\wr C_5$ x 2

Order 128: $C_2^4\times C_8$ x 2, $C_2^5\times C_4$

Order 64: $C_2^4\times C_4$ x 2, $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2\times C_4$

Order 5: $C_5$

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

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: $C_4\wr C_5.C_4$

Order 10240: $C_2\times C_4^4.C_{20}$, $C_4^4:C_{40}$

Order 5120: $C_4\wr C_5$ x 2, $C_2\times C_4^4:C_{10}$

Order 4096: $C_4^5.C_2^2$

Order 2560: $C_4^4:C_{10}$ x 2

Order 2048: $C_4^5.C_2$ x 2, $(C_2^2\times C_4^4).C_2$ x 2, $C_2\times C_4^5$

Order 1280: $C_2^6.C_{20}$, $C_4^4:C_5$

Order 1024: $(C_2\times C_4^4).C_2$ x 16, $(C_2^3\times C_4^3).C_2$ x 6, $C_2\times (C_2^2\times C_4^3).C_2$ x 4, $C_4^5$ x 4, $C_2^2\times C_4^4$ x 3

Order 640: $C_2^5:C_{20}$, $C_2^4:C_{40}$

Order 512: $C_2\times C_4^4$ x 41, $(C_2^2\times C_4^3).C_2$ x 22, $C_2\times (C_2\times C_4^3).C_2$ x 16, $C_2^3\times C_4^3$ x 11, $C_2^2\times (C_2^2\times C_4^2).C_2$ x 8, $(C_2^2\times C_4^3).C_2$ x 8

Order 320: $C_2^4:C_{20}$ x 2, $C_2^5:C_{10}$

Order 256: $C_2^2\times C_4^3$ x 151, $C_4^4$ x 58, $C_2^4.C_4^2$ x 48, $(C_2\times C_4^2):C_8$ x 16, $C_2^4.C_4^2$ x 16, $C_2^4\times C_4^2$ x 11, $C_2^4\times \OD_{16}$ x 2

Order 160: $C_2\wr C_5$ x 2

Order 128: $C_2^5\times C_4$, $C_2^4\times C_8$

Order 64: $C_2^4\times C_4$ x 2, $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2\times C_4$

Order 5: $C_5$

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

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20480: $C_4\wr C_5.C_4$ ($C_1$)

Order 10240: $C_4^4:C_{40}$ ($C_2$) x 2, $C_2\times C_4^4.C_{20}$ ($C_2$)

Order 5120: $C_4\wr C_5$ ($C_2^2$), $C_4\wr C_5$ ($C_4$), $C_2\times C_4^4:C_{10}$ ($C_4$)

Order 4096: $C_4^5.C_2^2$ ($C_5$)

Order 2560: $C_4^4:C_{10}$ ($C_2\times C_4$)

Order 2048: $C_4^5.C_2$ ($C_{10}$) x 2, $C_2\times C_4^5$ ($C_{10}$)

Order 1280: $C_4^4:C_5$ ($\OD_{16}$)

Order 1024: $C_4^5$ ($C_2\times C_{10}$), $C_4^5$ ($C_{20}$), $C_2^2\times C_4^4$ ($C_{20}$)

Order 512: $C_2\times C_4^4$ ($C_2\times C_{20}$)

Order 256: $C_4^4$ ($C_5\times \OD_{16}$), $C_2^4\times \OD_{16}$ ($C_2^4:C_5$)

Order 128: $C_2^4\times C_8$ ($C_2\wr C_5$) x 2, $C_2^5\times C_4$ ($C_2\wr C_5$)

Order 64: $C_2^6$ ($C_2^4:C_{20}$), $C_2^4\times C_4$ ($C_2^5:C_{10}$), $C_2^4\times C_4$ ($C_2^4:C_{20}$)

Order 32: $C_2^5$ ($C_2^5:C_{20}$)

Order 16: $C_2^4$ ($C_2^6.C_{20}$)

Order 8: $C_2\times C_4$ ($C_4^4:C_{10}$)

Order 4: $C_2^2$ ($C_4^4:C_{20}$), $C_4$ ($C_4^4:C_{20}$), $C_4$ ($C_2\times C_4^4:C_{10}$)

Order 2: $C_2$ ($C_4^4.C_{20}.C_2$)

Order 1: $C_1$ ($C_4\wr C_5.C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20480: $C_4\wr C_5.C_4$ ($C_1$)

Order 10240: $C_2\times C_4^4.C_{20}$ ($C_2$), $C_4^4:C_{40}$ ($C_2$)

Order 5120: $C_4\wr C_5$ ($C_2^2$), $C_4\wr C_5$ ($C_4$), $C_2\times C_4^4:C_{10}$ ($C_4$)

Order 4096: $C_4^5.C_2^2$ ($C_5$)

Order 2560: $C_4^4:C_{10}$ ($C_2\times C_4$)

Order 2048: $C_4^5.C_2$ ($C_{10}$), $C_2\times C_4^5$ ($C_{10}$)

Order 1280: $C_4^4:C_5$ ($\OD_{16}$)

Order 1024: $C_4^5$ ($C_2\times C_{10}$), $C_4^5$ ($C_{20}$), $C_2^2\times C_4^4$ ($C_{20}$)

Order 512: $C_2\times C_4^4$ ($C_2\times C_{20}$)

Order 256: $C_4^4$ ($C_5\times \OD_{16}$), $C_2^4\times \OD_{16}$ ($C_2^4:C_5$)

Order 128: $C_2^5\times C_4$ ($C_2\wr C_5$), $C_2^4\times C_8$ ($C_2\wr C_5$)

Order 64: $C_2^6$ ($C_2^4:C_{20}$), $C_2^4\times C_4$ ($C_2^5:C_{10}$), $C_2^4\times C_4$ ($C_2^4:C_{20}$)

Order 32: $C_2^5$ ($C_2^5:C_{20}$)

Order 16: $C_2^4$ ($C_2^6.C_{20}$)

Order 8: $C_2\times C_4$ ($C_4^4:C_{10}$)

Order 4: $C_2^2$ ($C_4^4:C_{20}$), $C_4$ ($C_4^4:C_{20}$), $C_4$ ($C_2\times C_4^4:C_{10}$)

Order 2: $C_2$ ($C_4^4.C_{20}.C_2$)

Order 1: $C_1$ ($C_4\wr C_5.C_4$)

Series

Derived series

$C_4\wr C_5.C_4$

$\rhd$

$C_2\times C_4^4$

$\rhd$

$C_1$

Chief series

$C_4\wr C_5.C_4$

$\rhd$

$C_4^4:C_{40}$

$\rhd$

$C_4\wr C_5$

$\rhd$

$C_4^5$

$\rhd$

$C_2\times C_4^4$

$\rhd$

$C_4^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_4\wr C_5.C_4$

$\rhd$

$C_2\times C_4^4$

$\rhd$

$C_4^4$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 12 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 $272 \times 272$ 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 $131 \times 131$ rational character table (warning: may be slow to load).