Abstract group 1600.1937: $C_{10}^2.D_8$

• Label: 1600.1937
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \cdot 5 \)
• Perm deg.: $26$
• Trans deg.: $1600$
• Rank: $3$

Group information

Description:	$C_{10}^2.D_8$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_{10}^2.C_2^3.C_2^6.S_5$, of order \(6144000\)\(\medspace = 2^{14} \cdot 3 \cdot 5^{3} \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	20	40
Elements	1	7	808	24	16	168	192	384	1600
Conjugacy classes	1	7	12	12	8	84	96	192	412
Divisions	1	7	8	6	2	42	24	12	102
Autjugacy classes	1	5	4	1	1	5	3	1	21

Dimension	1	2	4	8	16
Irr. complex chars.	16	396	0	0	0	412
Irr. rational chars.	4	10	28	36	24	102

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$1600$
Rank:	$3$
Inequivalent generating triples:	$336$

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:	$\langle a, b, c \mid a^{4}=b^{40}=c^{10}=[b,c]=1, b^{a}=b^{19}c^{5}, c^{a}=c^{9} \rangle$
Permutation group:	Degree $26$ $\langle(2,6)(3,4)(7,8)(9,10,11,12)(13,14,16,15)(18,20)(19,21)(23,24)(25,26), (1,2,3,7,5,8,4,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{10}.D_{20})$ $\,\rtimes\,$ $C_4$	$(C_{10}\times C_{40})$ $\,\rtimes\,$ $C_4$	$C_5$ $\,\rtimes\,$ $(C_{20}.C_4^2)$	$(C_2\times C_{40})$ $\,\rtimes\,$ $(C_5:C_4)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{10}^2$ . $D_8$	$C_{10}^2$ . $Q_{16}$	$C_{10}^2$ . $\SD_{16}$	$(C_{10}\times C_{20})$ . $Q_8$	all 61

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

Homology

Abelianization:	$C_{4}^{2} $
Schur multiplier:	$C_{2}^{2} \times C_{10}$
Commutator length:	$1$

Subgroups

There are 3384 subgroups in 456 conjugacy classes, 247 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_5:D_{20}$
Commutator:	$G' \simeq$ $C_5\times C_{20}$	$G/G' \simeq$ $C_4^2$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_5:D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{10}\times C_{40}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{10}^2.D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}^2$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.C_4^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 1600: $C_{10}^2.D_8$

Order 800: $C_{10}^2.D_4$ x 2, $C_2\times C_{10}\times C_{40}$

Order 400: $C_{10}.D_{20}$ x 6, $C_{10}\times C_{40}$ x 4, $C_{10}^2:C_4$ x 2, $C_2\times C_{10}\times C_{20}$

Order 320: $C_{20}.C_4^2$ x 6

Order 200: $C_{10}.D_{10}$ x 8, $C_{10}\times C_{20}$ x 6, $C_5\times C_{40}$ x 2, $C_2\times C_{10}^2$

Order 160: $C_2^2.D_{20}$ x 12, $C_2^2\times C_{40}$ x 6

Order 100: $C_{10}^2$ x 7, $C_5\times C_{20}$ x 4, $C_5^2:C_4$ x 4

Order 80: $C_{20}:C_4$ x 36, $C_2\times C_{40}$ x 24, $C_2^2.D_{10}$ x 12, $C_2^2\times C_{20}$ x 6

Order 64: $C_4.C_4^2$

Order 50: $C_5\times C_{10}$ x 7

Order 40: $C_{10}:C_4$ x 48, $C_2\times C_{20}$ x 36, $C_{40}$ x 12, $C_2^2\times C_{10}$ x 6

Order 32: $C_2^2.D_4$ x 2, $C_2^2\times C_8$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 42, $C_{20}$ x 24, $C_5:C_4$ x 24

Order 16: $C_4:C_4$ x 6, $C_2\times C_8$ x 4, $C_2^2\times C_4$ x 3

Order 10: $C_{10}$ x 42

Order 8: $C_2\times C_4$ x 14, $C_8$ x 2, $C_2^3$

Order 5: $C_5$ x 6

Order 4: $C_4$ x 8, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_{10}^2.D_8$

Order 800: $C_2\times C_{10}\times C_{40}$, $C_{10}^2.D_4$

Order 400: $C_{10}\times C_{40}$ x 2, $C_{10}.D_{20}$ x 2, $C_2\times C_{10}\times C_{20}$, $C_{10}^2:C_4$

Order 320: $C_{20}.C_4^2$

Order 200: $C_{10}\times C_{20}$ x 4, $C_{10}.D_{10}$ x 3, $C_2\times C_{10}^2$, $C_5\times C_{40}$

Order 160: $C_2^2\times C_{40}$, $C_2^2.D_{20}$

Order 100: $C_{10}^2$ x 5, $C_5\times C_{20}$ x 3, $C_5^2:C_4$

Order 80: $C_2\times C_{40}$ x 2, $C_{20}:C_4$ x 2, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_4.C_4^2$

Order 50: $C_5\times C_{10}$ x 5

Order 40: $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 3, $C_{40}$, $C_2^2\times C_{10}$

Order 32: $C_2^2\times C_8$, $C_2^2.D_4$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 5, $C_{20}$ x 3, $C_5:C_4$

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

Order 10: $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 7, $C_2^3$, $C_8$

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_{10}^2.D_8$ ($C_1$)

Order 800: $C_{10}^2.D_4$ ($C_2$) x 2, $C_2\times C_{10}\times C_{40}$ ($C_2$)

Order 400: $C_{10}.D_{20}$ ($C_4$) x 4, $C_{10}\times C_{40}$ ($C_4$) x 2, $C_2\times C_{10}\times C_{20}$ ($C_2^2$)

Order 200: $C_{10}\times C_{20}$ ($C_2\times C_4$) x 3, $C_{10}\times C_{20}$ ($D_4$) x 2, $C_2\times C_{10}^2$ ($D_4$), $C_{10}\times C_{20}$ ($Q_8$)

Order 160: $C_2^2\times C_{40}$ ($D_5$) x 6

Order 100: $C_5\times C_{20}$ ($C_4:C_4$) x 2, $C_{10}^2$ ($\SD_{16}$) x 2, $C_{10}^2$ ($C_2^2:C_4$) x 2, $C_5\times C_{20}$ ($C_2^2:C_4$), $C_5\times C_{20}$ ($C_4^2$), $C_{10}^2$ ($Q_{16}$), $C_{10}^2$ ($D_8$), $C_{10}^2$ ($C_4:C_4$)

Order 80: $C_2\times C_{40}$ ($C_5:C_4$) x 12, $C_2^2\times C_{20}$ ($D_{10}$) x 6

Order 50: $C_5\times C_{10}$ ($D_4:C_4$) x 2, $C_5\times C_{10}$ ($Q_8:C_4$) x 2, $C_5\times C_{10}$ ($C_2.C_4^2$), $C_5\times C_{10}$ ($C_8:C_4$), $C_5\times C_{10}$ ($C_8:C_4$)

Order 40: $C_2\times C_{20}$ ($C_5:D_4$) x 12, $C_2\times C_{20}$ ($C_4\times D_5$) x 12, $C_2\times C_{20}$ ($C_{10}:C_4$) x 6, $C_2\times C_{20}$ ($C_5:Q_8$) x 6, $C_2^2\times C_{10}$ ($D_{20}$) x 6

Order 32: $C_2^2\times C_8$ ($C_5:D_5$)

Order 25: $C_5^2$ ($C_4.C_4^2$)

Order 20: $C_2\times C_{10}$ ($C_{40}:C_2$) x 12, $C_2\times C_{10}$ ($D_{10}:C_4$) x 12, $C_{20}$ ($C_{10}.D_4$) x 12, $C_2\times C_{10}$ ($C_5:Q_{16}$) x 6, $C_2\times C_{10}$ ($D_{40}$) x 6, $C_2\times C_{10}$ ($C_{20}:C_4$) x 6, $C_{20}$ ($C_{10}.D_4$) x 6, $C_{20}$ ($C_{20}:C_4$) x 6

Order 16: $C_2\times C_8$ ($C_5^2:C_4$) x 2, $C_2^2\times C_4$ ($C_5:D_{10}$)

Order 10: $C_{10}$ ($D_{20}:C_4$) x 12, $C_{10}$ ($C_{10}.Q_{16}$) x 12, $C_{10}$ ($C_{10}.C_4^2$) x 6, $C_{10}$ ($C_{40}:C_4$) x 6, $C_{10}$ ($C_{40}:C_4$) x 6

Order 8: $C_2\times C_4$ ($C_{10}^2:C_2$) x 2, $C_2\times C_4$ ($C_{20}:D_5$) x 2, $C_2^3$ ($C_5:D_{20}$), $C_2\times C_4$ ($C_{10}.D_{10}$), $C_2\times C_4$ ($C_5^2:Q_8$)

Order 5: $C_5$ ($C_{20}.C_4^2$) x 6

Order 4: $C_2^2$ ($C_{40}:D_5$) x 2, $C_2^2$ ($C_{10}.D_{20}$) x 2, $C_4$ ($C_{10}^2.C_2^2$) x 2, $C_2^2$ ($C_5^2:Q_{16}$), $C_2^2$ ($C_5:D_{40}$), $C_2^2$ ($C_{10}.D_{20}$), $C_4$ ($C_5^2:C_4^2$), $C_4$ ($C_{10}^2:C_4$)

Order 2: $C_2$ ($C_{10}.D_{40}$) x 2, $C_2$ ($C_{10}^2.D_4$) x 2, $C_2$ ($C_{10}^2.D_4$), $C_2$ ($C_{10}.D_{40}$), $C_2$ ($C_{10}^2.D_4$)

Order 1: $C_1$ ($C_{10}^2.D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_{10}^2.D_8$ ($C_1$)

Order 800: $C_2\times C_{10}\times C_{40}$ ($C_2$), $C_{10}^2.D_4$ ($C_2$)

Order 400: $C_2\times C_{10}\times C_{20}$ ($C_2^2$), $C_{10}\times C_{40}$ ($C_4$), $C_{10}.D_{20}$ ($C_4$)

Order 200: $C_{10}\times C_{20}$ ($C_2\times C_4$) x 2, $C_2\times C_{10}^2$ ($D_4$), $C_{10}\times C_{20}$ ($Q_8$), $C_{10}\times C_{20}$ ($D_4$)

Order 160: $C_2^2\times C_{40}$ ($D_5$)

Order 100: $C_5\times C_{20}$ ($C_4:C_4$), $C_5\times C_{20}$ ($C_2^2:C_4$), $C_5\times C_{20}$ ($C_4^2$), $C_{10}^2$ ($Q_{16}$), $C_{10}^2$ ($\SD_{16}$), $C_{10}^2$ ($D_8$), $C_{10}^2$ ($C_4:C_4$), $C_{10}^2$ ($C_2^2:C_4$)

Order 80: $C_2^2\times C_{20}$ ($D_{10}$), $C_2\times C_{40}$ ($C_5:C_4$)

Order 50: $C_5\times C_{10}$ ($D_4:C_4$), $C_5\times C_{10}$ ($C_2.C_4^2$), $C_5\times C_{10}$ ($C_8:C_4$), $C_5\times C_{10}$ ($C_8:C_4$), $C_5\times C_{10}$ ($Q_8:C_4$)

Order 40: $C_2\times C_{20}$ ($C_5:D_4$), $C_2\times C_{20}$ ($C_{10}:C_4$), $C_2\times C_{20}$ ($C_4\times D_5$), $C_2\times C_{20}$ ($C_5:Q_8$), $C_2^2\times C_{10}$ ($D_{20}$)

Order 32: $C_2^2\times C_8$ ($C_5:D_5$)

Order 25: $C_5^2$ ($C_4.C_4^2$)

Order 20: $C_2\times C_{10}$ ($C_5:Q_{16}$), $C_2\times C_{10}$ ($D_{40}$), $C_2\times C_{10}$ ($C_{40}:C_2$), $C_2\times C_{10}$ ($D_{10}:C_4$), $C_2\times C_{10}$ ($C_{20}:C_4$), $C_{20}$ ($C_{10}.D_4$), $C_{20}$ ($C_{10}.D_4$), $C_{20}$ ($C_{20}:C_4$)

Order 16: $C_2\times C_8$ ($C_5^2:C_4$), $C_2^2\times C_4$ ($C_5:D_{10}$)

Order 10: $C_{10}$ ($C_{10}.C_4^2$), $C_{10}$ ($D_{20}:C_4$), $C_{10}$ ($C_{40}:C_4$), $C_{10}$ ($C_{40}:C_4$), $C_{10}$ ($C_{10}.Q_{16}$)

Order 8: $C_2^3$ ($C_5:D_{20}$), $C_2\times C_4$ ($C_{10}^2:C_2$), $C_2\times C_4$ ($C_{10}.D_{10}$), $C_2\times C_4$ ($C_{20}:D_5$), $C_2\times C_4$ ($C_5^2:Q_8$)

Order 5: $C_5$ ($C_{20}.C_4^2$)

Order 4: $C_2^2$ ($C_5^2:Q_{16}$), $C_2^2$ ($C_5:D_{40}$), $C_2^2$ ($C_{40}:D_5$), $C_2^2$ ($C_{10}.D_{20}$), $C_2^2$ ($C_{10}.D_{20}$), $C_4$ ($C_5^2:C_4^2$), $C_4$ ($C_{10}^2:C_4$), $C_4$ ($C_{10}^2.C_2^2$)

Order 2: $C_2$ ($C_{10}^2.D_4$), $C_2$ ($C_{10}.D_{40}$), $C_2$ ($C_{10}.D_{40}$), $C_2$ ($C_{10}^2.D_4$), $C_2$ ($C_{10}^2.D_4$)

Order 1: $C_1$ ($C_{10}^2.D_8$)

Series

Derived series

$C_{10}^2.D_8$

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_1$

Chief series

$C_{10}^2.D_8$

$\rhd$

$C_{10}^2.D_4$

$\rhd$

$C_2\times C_{10}\times C_{20}$

$\rhd$

$C_2\times C_{10}^2$

$\rhd$

$C_{10}^2$

$\rhd$

$C_5\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{10}^2.D_8$

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_5\times C_{10}$

$\rhd$

$C_5^2$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times C_8$

Character theory

Complex character table

See the $412 \times 412$ 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 $102 \times 102$ rational character table.