Abstract group 25000.ci: $C_5^4:(C_2\times C_{20})$

• Label: 25000.ci
• Order: \( 2^{3} \cdot 5^{5} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 5 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 5^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $25$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_5^4:(C_2\times C_{20})$
Order:	\(25000\)\(\medspace = 2^{3} \cdot 5^{5} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5^4.C_4^3.C_2^3.C_2^4$, of order \(5120000\)\(\medspace = 2^{13} \cdot 5^{4} \)
Composition factors:	$C_2$ x 3, $C_5$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20
Elements	1	675	2500	3124	8700	10000	25000
Conjugacy classes	1	3	4	424	72	16	520
Divisions	1	3	2	133	23	2	164
Autjugacy classes	1	2	1	11	6	1	22

Dimension	1	2	4	8	16	32
Irr. complex chars.	40	0	120	360	0	0	520
Irr. rational chars.	4	2	12	18	56	72	164

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	not computed	32
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{20}=b^{10}=c^{5}=d^{5}=e^{5}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $25$ $\langle(2,3,4,5)(6,8,10,7,9)(12,13,14,15)(16,18)(17,20,24,21)(19,23,22,25), (1,2)(3,4)(6,7,8,9,10)(11,12)(14,15)(16,17,19,22,24)(18,21,25,23,20)\rangle$
Transitive group:	40T14281				more information
Direct product:	$C_5$ $\, \times\, $ $(C_5^4:(C_2\times C_4))$
Semidirect product:	$C_5^3$ $\,\rtimes\,$ $(D_5:F_5)$	$(C_5^5:C_2)$ $\,\rtimes\,$ $C_4$	$C_5^4$ $\,\rtimes\,$ $(C_2\times F_5)$	$C_5^4$ $\,\rtimes\,$ $(C_2\times C_{20})$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_5^5$ . $(C_2\times C_4)$	$(C_5^5:C_2)$ . $C_2^2$	$(C_5^4:C_2^2)$ . $C_{10}$	$(C_5\times C_5^4:C_2^2)$ . $C_2$	all 5
Aut. group:	$\Aut(C_2^8:A_4)$

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:	$C_{5} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 152864 subgroups in 7024 conjugacy classes, 58 normal (10 characteristic).

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

Special subgroups

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

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 25000: $C_5^4:(C_2\times C_{20})$

Order 12500: $C_5\times C_5^4:C_4$ x 2, $C_5\times C_5^4:C_2^2$

Order 6250: $C_5^5:C_2$ x 2, $C_5^5:C_2$

Order 5000: $C_5^3:(C_2\times C_{20})$ x 4, $C_5^4:(C_2\times C_4)$

Order 3125: $C_5^5$

Order 2500: $C_5\times C_5:(C_5:F_5)$ x 16, $C_5\times D_5\times C_5:D_5$ x 8, $C_5^4:C_4$ x 2, $C_5^4:C_2^2$

Order 1250: $C_5^3:C_{10}$ x 48, $C_5^2\wr C_2$ x 22, $D_5\times C_5^3$ x 8, $C_5^4:C_2$

Order 1000: $C_5^3:(C_2\times C_4)$ x 4, $D_5^2.C_{10}$ x 4, $C_5\times C_{10}:F_5$ x 2

Order 625: $C_5^4$ x 133

Order 500: $C_5^2:C_{20}$ x 40, $C_5\times D_5^2$ x 20, $C_5^3:C_4$ x 16, $C_5:D_5^2$ x 8, $C_5^2:C_{20}$ x 4, $C_5^2:D_{10}$ x 2

Order 250: $C_5^2:C_{10}$ x 252, $D_5\times C_5^2$ x 120, $C_5^3:C_2$ x 48, $C_5^2\times C_{10}$ x 2

Order 200: $C_{10}\times F_5$ x 4, $D_5:F_5$ x 4, $C_{10}:F_5$ x 2

Order 125: $C_5^3$ x 2738

Order 100: $C_5:F_5$ x 40, $D_5^2$ x 20, $C_5\times F_5$ x 16, $C_5\times D_{10}$ x 8, $C_5:F_5$ x 4, $C_5:D_{10}$ x 2

Order 50: $C_5:D_5$ x 232, $C_5\times D_5$ x 168, $C_5\times C_{10}$ x 22

Order 40: $C_2\times F_5$ x 4, $C_2\times C_{20}$

Order 25: $C_5^2$ x 2738

Order 20: $F_5$ x 16, $D_{10}$ x 8, $C_{20}$ x 2, $C_2\times C_{10}$

Order 10: $D_5$ x 56, $C_{10}$ x 23

Order 8: $C_2\times C_4$

Order 5: $C_5$ x 133

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 25000: $C_5^4:(C_2\times C_{20})$

Order 12500: $C_5\times C_5^4:C_2^2$, $C_5\times C_5^4:C_4$

Order 6250: $C_5^5:C_2$, $C_5^5:C_2$

Order 5000: $C_5^3:(C_2\times C_{20})$, $C_5^4:(C_2\times C_4)$

Order 3125: $C_5^5$

Order 2500: $C_5\times C_5:(C_5:F_5)$ x 2, $C_5\times D_5\times C_5:D_5$ x 2, $C_5^4:C_2^2$, $C_5^4:C_4$

Order 1250: $C_5^3:C_{10}$ x 5, $C_5^2\wr C_2$ x 5, $D_5\times C_5^3$ x 2, $C_5^4:C_2$

Order 1000: $C_5\times C_{10}:F_5$, $C_5^3:(C_2\times C_4)$, $D_5^2.C_{10}$

Order 625: $C_5^4$ x 11

Order 500: $C_5^2:C_{20}$ x 4, $C_5\times D_5^2$ x 3, $C_5:D_5^2$ x 2, $C_5^3:C_4$ x 2, $C_5^2:D_{10}$, $C_5^2:C_{20}$

Order 250: $C_5^2:C_{10}$ x 17, $D_5\times C_5^2$ x 10, $C_5^3:C_2$ x 5, $C_5^2\times C_{10}$

Order 200: $C_{10}:F_5$, $C_{10}\times F_5$, $D_5:F_5$

Order 125: $C_5^3$ x 50

Order 100: $C_5:F_5$ x 4, $D_5^2$ x 3, $C_5\times F_5$ x 2, $C_5\times D_{10}$ x 2, $C_5:D_{10}$, $C_5:F_5$

Order 50: $C_5\times D_5$ x 15, $C_5:D_5$ x 13, $C_5\times C_{10}$ x 5

Order 40: $C_2\times C_{20}$, $C_2\times F_5$

Order 25: $C_5^2$ x 50

Order 20: $D_{10}$ x 2, $F_5$ x 2, $C_2\times C_{10}$, $C_{20}$

Order 10: $D_5$ x 7, $C_{10}$ x 6

Order 8: $C_2\times C_4$

Order 5: $C_5$ x 11

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 25000: $C_5^4:(C_2\times C_{20})$ ($C_1$)

Order 12500: $C_5\times C_5^4:C_4$ ($C_2$) x 2, $C_5\times C_5^4:C_2^2$ ($C_2$)

Order 6250: $C_5^5:C_2$ ($C_4$) x 2, $C_5^5:C_2$ ($C_2^2$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_5$)

Order 3125: $C_5^5$ ($C_2\times C_4$)

Order 2500: $C_5^4:C_4$ ($C_{10}$) x 2, $C_5^4:C_2^2$ ($C_{10}$)

Order 1250: $C_5^2\wr C_2$ ($F_5$) x 4, $C_5^2\wr C_2$ ($C_{20}$) x 2, $C_5^4:C_2$ ($C_2\times C_{10}$)

Order 625: $C_5^4$ ($C_2\times F_5$) x 4, $C_5^4$ ($C_2\times C_{20}$)

Order 250: $C_5^2:C_{10}$ ($C_5\times F_5$) x 4, $C_5^2:C_{10}$ ($C_5:F_5$) x 2

Order 125: $C_5^3$ ($C_{10}\times F_5$) x 4, $C_5^3$ ($D_5:F_5$) x 4, $C_5^3$ ($C_{10}:F_5$) x 2

Order 50: $C_5:D_5$ ($C_5^2:C_{20}$) x 2

Order 25: $C_5^2$ ($C_5^3:(C_2\times C_4)$) x 4, $C_5^2$ ($D_5^2.C_{10}$) x 4, $C_5^2$ ($C_5\times C_{10}:F_5$) x 2

Order 5: $C_5$ ($C_5^3:(C_2\times C_{20})$) x 4, $C_5$ ($C_5^4:(C_2\times C_4)$)

Order 1: $C_1$ ($C_5^4:(C_2\times C_{20})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 25000: $C_5^4:(C_2\times C_{20})$ ($C_1$)

Order 12500: $C_5\times C_5^4:C_2^2$ ($C_2$), $C_5\times C_5^4:C_4$ ($C_2$)

Order 6250: $C_5^5:C_2$ ($C_4$), $C_5^5:C_2$ ($C_2^2$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_5$)

Order 3125: $C_5^5$ ($C_2\times C_4$)

Order 2500: $C_5^4:C_2^2$ ($C_{10}$), $C_5^4:C_4$ ($C_{10}$)

Order 1250: $C_5^4:C_2$ ($C_2\times C_{10}$), $C_5^2\wr C_2$ ($F_5$), $C_5^2\wr C_2$ ($C_{20}$)

Order 625: $C_5^4$ ($C_2\times C_{20}$), $C_5^4$ ($C_2\times F_5$)

Order 250: $C_5^2:C_{10}$ ($C_5\times F_5$), $C_5^2:C_{10}$ ($C_5:F_5$)

Order 125: $C_5^3$ ($C_{10}:F_5$), $C_5^3$ ($C_{10}\times F_5$), $C_5^3$ ($D_5:F_5$)

Order 50: $C_5:D_5$ ($C_5^2:C_{20}$)

Order 25: $C_5^2$ ($C_5\times C_{10}:F_5$), $C_5^2$ ($C_5^3:(C_2\times C_4)$), $C_5^2$ ($D_5^2.C_{10}$)

Order 5: $C_5$ ($C_5^3:(C_2\times C_{20})$), $C_5$ ($C_5^4:(C_2\times C_4)$)

Order 1: $C_1$ ($C_5^4:(C_2\times C_{20})$)

Series

Derived series

$C_5^4:(C_2\times C_{20})$

$\rhd$

$C_5^4$

$\rhd$

$C_1$

Chief series

$C_5^4:(C_2\times C_{20})$

$\rhd$

$C_5\times C_5^4:C_2^2$

$\rhd$

$C_5^5:C_2$

$\rhd$

$C_5^2\wr C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^3$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5^4:(C_2\times C_{20})$

$\rhd$

$C_5^4$

Upper central series

$C_1$

$\lhd$

$C_5$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $520 \times 520$ character table is not available for this group.

Rational character table

See the $164 \times 164$ rational character table (warning: may be slow to load).