Abstract group 250000.bm: $C_5^6:C_2^4$

• Label: 250000.bm
• Order: \( 2^{4} \cdot 5^{6} \)
• Exponent: \( 2 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3 \cdot 5^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$C_5^6:C_2^4$
Order:	\(250000\)\(\medspace = 2^{4} \cdot 5^{6} \)
Exponent:	\(10\)\(\medspace = 2 \cdot 5 \)
Automorphism group:	$C_5^6:(D_4^3.C_4\wr S_3)$, of order \(3072000000\)\(\medspace = 2^{16} \cdot 3 \cdot 5^{6} \)
Composition factors:	$C_2$ x 4, $C_5$ x 6
Derived length:	$2$

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

Group statistics

Order	1	2	5	10
Elements	1	18575	15624	215800	250000
Conjugacy classes	1	15	1160	520	1696
Divisions	1	15	580	260	856
Autjugacy classes	1	4	9	10	24

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{10}=c^{10}=d^{10}=e^{5}=f^{5}=g^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,2,6,3,7,10,4,8,9,5)(11,12)(13,14)(16,17,20,19,23)(26,27,30,29,28), (1,5,6,8,7,10,4,3,9,2) \!\cdots\! \rangle$
Transitive group:	40T106938				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:	$C_5^2$ . $D_5^4$	$C_5^4$ . $D_{10}^2$ (2)	$C_5$ . $(C_5:D_5^4)$	$(C_5^2:D_5^3)$ . $D_5$	all 23

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 303 normal subgroups (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_5^6:C_2^4$
Commutator:	$G' \simeq$ $C_5^6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^6:C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^6$	$G/\operatorname{Fit} \simeq$ $C_2^4$
Radical:	$R \simeq$ $C_5^6:C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^6$

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 25 (order at least 10000) are shown, as well as all normal subgroups of any index.

Order 250000: $C_5^6:C_2^4$

Order 125000: $C_5^6.C_2^3$ x 6, $C_5^6:C_2^3$ x 6, $C_5^6:C_2^3$ x 2, $C_5:D_5\times C_5^4:C_2^2$

Order 62500: $C_5\times C_5^5:C_2^2$ x 12, $C_5\times C_5^5:C_2^2$ x 12, $C_5^5.D_{10}$ x 4, $C_5\times C_5:D_5\times C_5\times C_5:D_5$ x 3, $C_5^5.D_{10}$ x 3, $C_5^5.D_{10}$

Order 50000: $C_5:D_5^4$ x 6

Order 31250: $C_5^2\times C_5^4:C_2$ x 8, $C_5\times C_5^5:C_2$ x 3, $C_5^4\times C_5:D_5$ x 3, $C_5^6:C_2$

Order 25000: $C_5^5.C_2^3$ x 48, $C_5^2:D_5^3$ x 24, $C_5^2:D_5^3$ x 18, $C_5^5.C_2^3$ x 12, $C_5:D_5\times D_5\times C_5:D_5$ x 12, $C_5^2:D_5^3$ x 6

Order 15625: $C_5^6$

Order 12500: $C_5^5:C_2^2$ x 168, $C_5^5:C_2^2$ x 72, $C_5\times C_5^4:C_2^2$ x 72, $C_5^2:D_5\times C_5:D_5$ x 48, $C_5:D_5\times C_5^2:D_5$ x 32, $C_5^5:C_2^2$ x 24, $C_5^2\times C_5:D_5\times D_5$ x 24, $C_5\times C_5^4:C_2^2$ x 24, $C_5^5:C_2^2$ x 24, $C_5^2\times C_5^3:C_2^2$ x 24, $C_5\times C_5^4:C_2^2$ x 24, $C_5^4:C_2\times D_5$ x 12

Order 10000: $D_5^4$ x 12, $C_{10}:D_5^3$ x 3

Order 6250: $C_5^5:C_2$ x 24, $C_5^5:C_2$ x 12, $C_5^5:C_2$ x 6

Order 3125: $C_5^5$ x 6

Order 2500: $C_5^4:C_2^2$ x 12, $(C_5:D_5)^2$ x 3

Order 1250: $C_5^3:C_{10}$ x 24, $C_5^2\wr C_2$ x 18, $C_5^4:C_2$ x 3

Order 625: $C_5^4$ x 15

Order 250: $C_5^2:C_{10}$ x 12, $C_5^3:C_2$ x 8

Order 125: $C_5^3$ x 20

Order 50: $C_5:D_5$ x 3

Order 25: $C_5^2$ x 15

Order 16: $C_2^4$

Order 5: $C_5$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 25 (order at least 10000) are shown, as well as all normal subgroups of any index.

Order 250000: $C_5^6:C_2^4$

Order 125000: $C_5:D_5\times C_5^4:C_2^2$, $C_5^6.C_2^3$, $C_5^6:C_2^3$, $C_5^6:C_2^3$

Order 62500: $C_5\times C_5^5:C_2^2$, $C_5\times C_5:D_5\times C_5\times C_5:D_5$, $C_5^5.D_{10}$, $C_5^5.D_{10}$, $C_5^5.D_{10}$, $C_5\times C_5^5:C_2^2$

Order 50000: $C_5:D_5^4$

Order 31250: $C_5\times C_5^5:C_2$, $C_5^2\times C_5^4:C_2$, $C_5^4\times C_5:D_5$, $C_5^6:C_2$

Order 25000: $C_5:D_5\times D_5\times C_5:D_5$ x 2, $C_5^5.C_2^3$ x 2, $C_5^2:D_5^3$ x 2, $C_5^5.C_2^3$, $C_5^2:D_5^3$, $C_5^2:D_5^3$

Order 15625: $C_5^6$

Order 12500: $C_5^5:C_2^2$ x 6, $C_5^5:C_2^2$ x 3, $C_5^5:C_2^2$ x 3, $C_5^2\times C_5:D_5\times D_5$ x 2, $C_5^4:C_2\times D_5$ x 2, $C_5^5:C_2^2$ x 2, $C_5\times C_5^4:C_2^2$ x 2, $C_5\times C_5^4:C_2^2$, $C_5^2:D_5\times C_5:D_5$, $C_5:D_5\times C_5^2:D_5$, $C_5^2\times C_5^3:C_2^2$, $C_5\times C_5^4:C_2^2$

Order 10000: $D_5^4$, $C_{10}:D_5^3$

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

Order 3125: $C_5^5$

Order 2500: $(C_5:D_5)^2$, $C_5^4:C_2^2$

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

Order 625: $C_5^4$ x 2

Order 250: $C_5^3:C_2$, $C_5^2:C_{10}$

Order 125: $C_5^3$ x 2

Order 50: $C_5:D_5$

Order 25: $C_5^2$ x 2

Order 16: $C_2^4$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 250000: $C_5^6:C_2^4$ ($C_1$)

Order 125000: $C_5^6.C_2^3$ ($C_2$) x 6, $C_5^6:C_2^3$ ($C_2$) x 6, $C_5^6:C_2^3$ ($C_2$) x 2, $C_5:D_5\times C_5^4:C_2^2$ ($C_2$)

Order 62500: $C_5\times C_5^5:C_2^2$ ($C_2^2$) x 12, $C_5\times C_5^5:C_2^2$ ($C_2^2$) x 12, $C_5^5.D_{10}$ ($C_2^2$) x 4, $C_5\times C_5:D_5\times C_5\times C_5:D_5$ ($C_2^2$) x 3, $C_5^5.D_{10}$ ($C_2^2$) x 3, $C_5^5.D_{10}$ ($C_2^2$)

Order 31250: $C_5^2\times C_5^4:C_2$ ($C_2^3$) x 8, $C_5\times C_5^5:C_2$ ($C_2^3$) x 3, $C_5^4\times C_5:D_5$ ($C_2^3$) x 3, $C_5^6:C_2$ ($C_2^3$)

Order 25000: $C_5^2:D_5^3$ ($D_5$) x 6

Order 15625: $C_5^6$ ($C_2^4$)

Order 12500: $C_5\times C_5^4:C_2^2$ ($D_{10}$) x 24, $C_5^5:C_2^2$ ($D_{10}$) x 12, $C_5^5:C_2^2$ ($D_{10}$) x 6

Order 6250: $C_5^5:C_2$ ($C_2\times D_{10}$) x 24, $C_5^5:C_2$ ($C_2\times D_{10}$) x 12, $C_5^5:C_2$ ($C_2\times D_{10}$) x 6

Order 3125: $C_5^5$ ($C_2^2\times D_{10}$) x 6

Order 2500: $C_5^4:C_2^2$ ($D_5^2$) x 12, $(C_5:D_5)^2$ ($D_5^2$) x 3

Order 1250: $C_5^3:C_{10}$ ($D_5\times D_{10}$) x 24, $C_5^2\wr C_2$ ($D_5\times D_{10}$) x 18, $C_5^4:C_2$ ($D_5\times D_{10}$) x 3

Order 625: $C_5^4$ ($D_{10}^2$) x 15

Order 250: $C_5^2:C_{10}$ ($D_5^3$) x 12, $C_5^3:C_2$ ($D_5^3$) x 8

Order 125: $C_5^3$ ($C_2\times D_5^3$) x 20

Order 50: $C_5:D_5$ ($C_5:D_5^3$) x 3

Order 25: $C_5^2$ ($D_5^4$) x 12, $C_5^2$ ($C_{10}:D_5^3$) x 3

Order 5: $C_5$ ($C_5:D_5^4$) x 6

Order 1: $C_1$ ($C_5^6:C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 250000: $C_5^6:C_2^4$ ($C_1$)

Order 125000: $C_5:D_5\times C_5^4:C_2^2$ ($C_2$), $C_5^6.C_2^3$ ($C_2$), $C_5^6:C_2^3$ ($C_2$), $C_5^6:C_2^3$ ($C_2$)

Order 62500: $C_5\times C_5^5:C_2^2$ ($C_2^2$), $C_5\times C_5:D_5\times C_5\times C_5:D_5$ ($C_2^2$), $C_5^5.D_{10}$ ($C_2^2$), $C_5^5.D_{10}$ ($C_2^2$), $C_5^5.D_{10}$ ($C_2^2$), $C_5\times C_5^5:C_2^2$ ($C_2^2$)

Order 31250: $C_5\times C_5^5:C_2$ ($C_2^3$), $C_5^2\times C_5^4:C_2$ ($C_2^3$), $C_5^4\times C_5:D_5$ ($C_2^3$), $C_5^6:C_2$ ($C_2^3$)

Order 25000: $C_5^2:D_5^3$ ($D_5$)

Order 15625: $C_5^6$ ($C_2^4$)

Order 12500: $C_5^5:C_2^2$ ($D_{10}$), $C_5\times C_5^4:C_2^2$ ($D_{10}$), $C_5^5:C_2^2$ ($D_{10}$)

Order 6250: $C_5^5:C_2$ ($C_2\times D_{10}$), $C_5^5:C_2$ ($C_2\times D_{10}$), $C_5^5:C_2$ ($C_2\times D_{10}$)

Order 3125: $C_5^5$ ($C_2^2\times D_{10}$)

Order 2500: $(C_5:D_5)^2$ ($D_5^2$), $C_5^4:C_2^2$ ($D_5^2$)

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

Order 625: $C_5^4$ ($D_{10}^2$) x 2

Order 250: $C_5^3:C_2$ ($D_5^3$), $C_5^2:C_{10}$ ($D_5^3$)

Order 125: $C_5^3$ ($C_2\times D_5^3$) x 2

Order 50: $C_5:D_5$ ($C_5:D_5^3$)

Order 25: $C_5^2$ ($D_5^4$), $C_5^2$ ($C_{10}:D_5^3$)

Order 5: $C_5$ ($C_5:D_5^4$)

Order 1: $C_1$ ($C_5^6:C_2^4$)

Series

Derived series

$C_5^6:C_2^4$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Chief series

$C_5^6:C_2^4$

$\rhd$

$C_5^6:C_2^3$

$\rhd$

$C_5\times C_5:D_5\times C_5\times C_5:D_5$

$\rhd$

$C_5^5:C_2^2$

$\rhd$

$C_5^5:C_2$

$\rhd$

$C_5^2\wr C_2$

$\rhd$

$C_5^4$

$\rhd$

$C_5^3$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_5^6:C_2^4$

$\rhd$

$C_5^6$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

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

Character theory

Complex character table

The $1696 \times 1696$ character table is not available for this group.

Rational character table

The $856 \times 856$ rational character table is not available for this group.