Abstract group 1600.1455: $C_{100}.C_2^4$

• Label: 1600.1455
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \cdot 5^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \cdot 5 \cdot 7 \)
• Perm deg.: $39$
• Trans deg.: $1600$
• Rank: $4$

Group information

Description:	$C_{100}.C_2^4$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(200\)\(\medspace = 2^{3} \cdot 5^{2} \)
Automorphism group:	$C_{50}.C_{20}.C_2^6.C_2.\PSL(2,7)$, of order \(21504000\)\(\medspace = 2^{13} \cdot 3 \cdot 5^{3} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	4	5	8	10	20	25	50	100
Elements	1	15	16	4	800	60	64	20	300	320	1600
Conjugacy classes	1	15	16	2	32	30	32	10	150	160	448
Divisions	1	15	8	1	8	15	8	1	15	8	80
Autjugacy classes	1	2	2	1	1	2	2	1	2	2	16

Dimension	1	2	4	8	20	40
Irr. complex chars.	64	384	0	0	0	0	448
Irr. rational chars.	16	8	24	8	16	8	80

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$1600$
Rank:	$4$
Inequivalent generating quadruples:	$93000$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{8}=d^{50}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=1, d^{c}=d^{49} \rangle$
Permutation group:	Degree $39$ $\langle(1,2)(3,4)(7,8,10,12,9,11,13,14)(16,18)(17,21)(19,24)(20,27)(22,30)(23,29) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $(C_{25}:C_8)$
Semidirect product:	$(C_2^2\times C_{50})$ $\,\rtimes\,$ $C_8$	$C_{50}$ $\,\rtimes\,$ $(C_2^2\times C_8)$	$C_{25}$ $\,\rtimes\,$ $(C_2^3\times C_8)$	$(C_2\times C_{50})$ $\,\rtimes\,$ $(C_2\times C_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{100}$ . $C_2^4$	$(C_2^3\times C_{20})$ . $D_5$	$(C_2^3\times C_4)$ . $D_{25}$	$(C_2^2\times C_4)$ . $D_{50}$	all 37

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

Homology

Abelianization:	$C_{2}^{3} \times C_{8} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1935 subgroups in 507 conjugacy classes, 405 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_4$	$G/Z \simeq$ $D_{25}$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_2^3\times C_8$
Frattini:	$\Phi \simeq$ $C_{20}$	$G/\Phi \simeq$ $C_2^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_{100}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{100}.C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_5:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1600: $C_{100}.C_2^4$

Order 800: $C_{100}.C_2^3$ x 14, $C_2^3\times C_{100}$

Order 400: $C_{50}:C_8$ x 28, $C_2^2\times C_{100}$ x 14, $C_2^3\times C_{50}$

Order 320: $C_{20}.C_2^4$

Order 200: $C_2\times C_{100}$ x 28, $C_2^2\times C_{50}$ x 15, $C_{25}:C_8$ x 8

Order 160: $C_{20}.C_2^3$ x 14, $C_2^3\times C_{20}$

Order 100: $C_2\times C_{50}$ x 35, $C_{100}$ x 8

Order 80: $C_{10}:C_8$ x 28, $C_2^2\times C_{20}$ x 14, $C_2^3\times C_{10}$

Order 64: $C_2^3\times C_8$

Order 50: $C_{50}$ x 15

Order 40: $C_2\times C_{20}$ x 28, $C_2^2\times C_{10}$ x 15, $C_5:C_8$ x 8

Order 32: $C_2^2\times C_8$ x 14, $C_2^3\times C_4$

Order 25: $C_{25}$

Order 20: $C_2\times C_{10}$ x 35, $C_{20}$ x 8

Order 16: $C_2\times C_8$ x 28, $C_2^2\times C_4$ x 14, $C_2^4$

Order 10: $C_{10}$ x 15

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15, $C_8$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_{100}.C_2^4$

Order 800: $C_2^3\times C_{100}$, $C_{100}.C_2^3$

Order 400: $C_2^2\times C_{100}$ x 2, $C_{50}:C_8$, $C_2^3\times C_{50}$

Order 320: $C_{20}.C_2^4$

Order 200: $C_2\times C_{100}$ x 2, $C_2^2\times C_{50}$ x 2, $C_{25}:C_8$

Order 160: $C_2^3\times C_{20}$, $C_{20}.C_2^3$

Order 100: $C_2\times C_{50}$ x 2, $C_{100}$ x 2

Order 80: $C_2^2\times C_{20}$ x 2, $C_{10}:C_8$, $C_2^3\times C_{10}$

Order 64: $C_2^3\times C_8$

Order 50: $C_{50}$ x 2

Order 40: $C_2\times C_{20}$ x 2, $C_2^2\times C_{10}$ x 2, $C_5:C_8$

Order 32: $C_2^3\times C_4$, $C_2^2\times C_8$

Order 25: $C_{25}$

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

Order 16: $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^4$

Order 10: $C_{10}$ x 2

Order 8: $C_2^3$ x 2, $C_2\times C_4$ x 2, $C_8$

Order 5: $C_5$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_{100}.C_2^4$ ($C_1$)

Order 800: $C_{100}.C_2^3$ ($C_2$) x 14, $C_2^3\times C_{100}$ ($C_2$)

Order 400: $C_{50}:C_8$ ($C_2^2$) x 28, $C_2^2\times C_{100}$ ($C_2^2$) x 7, $C_2^2\times C_{100}$ ($C_4$) x 7, $C_2^3\times C_{50}$ ($C_4$)

Order 200: $C_2\times C_{100}$ ($C_2\times C_4$) x 21, $C_2^2\times C_{50}$ ($C_8$) x 8, $C_{25}:C_8$ ($C_2^3$) x 8, $C_2\times C_{100}$ ($C_2^3$) x 7, $C_2^2\times C_{50}$ ($C_2\times C_4$) x 7

Order 160: $C_2^3\times C_{20}$ ($D_5$)

Order 100: $C_2\times C_{50}$ ($C_2\times C_8$) x 28, $C_2\times C_{50}$ ($C_2^2\times C_4$) x 7, $C_{100}$ ($C_2^2\times C_4$) x 7, $C_{100}$ ($C_2^4$)

Order 80: $C_2^2\times C_{20}$ ($D_{10}$) x 7, $C_2^2\times C_{20}$ ($C_5:C_4$) x 7, $C_2^3\times C_{10}$ ($C_5:C_4$)

Order 50: $C_{50}$ ($C_2^2\times C_8$) x 14, $C_{50}$ ($C_2^3\times C_4$)

Order 40: $C_2\times C_{20}$ ($C_{10}:C_4$) x 21, $C_2^2\times C_{10}$ ($C_5:C_8$) x 8, $C_2\times C_{20}$ ($C_2\times D_{10}$) x 7, $C_2^2\times C_{10}$ ($C_{10}:C_4$) x 7

Order 32: $C_2^3\times C_4$ ($D_{25}$)

Order 25: $C_{25}$ ($C_2^3\times C_8$)

Order 20: $C_2\times C_{10}$ ($C_{10}:C_8$) x 28, $C_2\times C_{10}$ ($C_2^2.D_{10}$) x 7, $C_{20}$ ($C_2^2.D_{10}$) x 7, $C_{20}$ ($C_2^2\times D_{10}$)

Order 16: $C_2^2\times C_4$ ($D_{50}$) x 7, $C_2^2\times C_4$ ($C_{25}:C_4$) x 7, $C_2^4$ ($C_{25}:C_4$)

Order 10: $C_{10}$ ($C_{20}.C_2^3$) x 14, $C_{10}$ ($C_{10}.C_2^4$)

Order 8: $C_2\times C_4$ ($C_{50}:C_4$) x 21, $C_2^3$ ($C_{25}:C_8$) x 8, $C_2^3$ ($C_{50}:C_4$) x 7, $C_2\times C_4$ ($C_2\times D_{50}$) x 7

Order 5: $C_5$ ($C_{20}.C_2^4$)

Order 4: $C_2^2$ ($C_{50}:C_8$) x 28, $C_2^2$ ($C_2^2.D_{50}$) x 7, $C_4$ ($C_2^2.D_{50}$) x 7, $C_4$ ($C_2^2\times D_{50}$)

Order 2: $C_2$ ($C_{100}.C_2^3$) x 14, $C_2$ ($C_2^3.D_{50}$)

Order 1: $C_1$ ($C_{100}.C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_{100}.C_2^4$ ($C_1$)

Order 800: $C_2^3\times C_{100}$ ($C_2$), $C_{100}.C_2^3$ ($C_2$)

Order 400: $C_{50}:C_8$ ($C_2^2$), $C_2^3\times C_{50}$ ($C_4$), $C_2^2\times C_{100}$ ($C_2^2$), $C_2^2\times C_{100}$ ($C_4$)

Order 200: $C_2\times C_{100}$ ($C_2^3$), $C_2\times C_{100}$ ($C_2\times C_4$), $C_2^2\times C_{50}$ ($C_2\times C_4$), $C_2^2\times C_{50}$ ($C_8$), $C_{25}:C_8$ ($C_2^3$)

Order 160: $C_2^3\times C_{20}$ ($D_5$)

Order 100: $C_2\times C_{50}$ ($C_2\times C_8$), $C_2\times C_{50}$ ($C_2^2\times C_4$), $C_{100}$ ($C_2^4$), $C_{100}$ ($C_2^2\times C_4$)

Order 80: $C_2^3\times C_{10}$ ($C_5:C_4$), $C_2^2\times C_{20}$ ($D_{10}$), $C_2^2\times C_{20}$ ($C_5:C_4$)

Order 50: $C_{50}$ ($C_2^3\times C_4$), $C_{50}$ ($C_2^2\times C_8$)

Order 40: $C_2\times C_{20}$ ($C_{10}:C_4$), $C_2\times C_{20}$ ($C_2\times D_{10}$), $C_2^2\times C_{10}$ ($C_{10}:C_4$), $C_2^2\times C_{10}$ ($C_5:C_8$)

Order 32: $C_2^3\times C_4$ ($D_{25}$)

Order 25: $C_{25}$ ($C_2^3\times C_8$)

Order 20: $C_2\times C_{10}$ ($C_{10}:C_8$), $C_2\times C_{10}$ ($C_2^2.D_{10}$), $C_{20}$ ($C_2^2\times D_{10}$), $C_{20}$ ($C_2^2.D_{10}$)

Order 16: $C_2^4$ ($C_{25}:C_4$), $C_2^2\times C_4$ ($D_{50}$), $C_2^2\times C_4$ ($C_{25}:C_4$)

Order 10: $C_{10}$ ($C_{10}.C_2^4$), $C_{10}$ ($C_{20}.C_2^3$)

Order 8: $C_2^3$ ($C_{50}:C_4$), $C_2^3$ ($C_{25}:C_8$), $C_2\times C_4$ ($C_{50}:C_4$), $C_2\times C_4$ ($C_2\times D_{50}$)

Order 5: $C_5$ ($C_{20}.C_2^4$)

Order 4: $C_2^2$ ($C_{50}:C_8$), $C_2^2$ ($C_2^2.D_{50}$), $C_4$ ($C_2^2\times D_{50}$), $C_4$ ($C_2^2.D_{50}$)

Order 2: $C_2$ ($C_2^3.D_{50}$), $C_2$ ($C_{100}.C_2^3$)

Order 1: $C_1$ ($C_{100}.C_2^4$)

Series

Derived series

$C_{100}.C_2^4$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$C_{100}.C_2^4$

$\rhd$

$C_{100}.C_2^3$

$\rhd$

$C_2^2\times C_{100}$

$\rhd$

$C_2\times C_{100}$

$\rhd$

$C_{100}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{100}.C_2^4$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_4$

Character theory

Complex character table

See the $448 \times 448$ 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 $80 \times 80$ rational character table.