Abstract group 1600.4623: $C_{20}^2.C_2^2$

• Label: 1600.4623
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $22$
• Trans deg.: $160$
• Rank: $3$

Group information

Description:	$C_{20}^2.C_2^2$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_5:(C_2^6.C_2^6)$, of order \(20480\)\(\medspace = 2^{12} \cdot 5 \)
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	11	20	24	160	264	480	640	1600
Conjugacy classes	1	7	12	14	20	98	168	80	400
Divisions	1	7	7	4	5	28	28	5	85
Autjugacy classes	1	4	5	3	3	12	15	3	46

Dimension	1	2	4	8	16	32
Irr. complex chars.	160	200	40	0	0	0	400
Irr. rational chars.	8	6	21	29	18	3	85

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$160$
Rank:	$3$
Inequivalent generating triples:	$62496$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{8}=b^{10}=c^{20}=[a,b]=1, c^{a}=c^{9}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $22$ $\langle(1,2)(3,4)(11,12)(13,14)(15,16,18,20,17,19,21,22), (3,4)(15,17)(16,19)(18,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 50 & 6 \\ 3 & 50 \end{array}\right), \left(\begin{array}{rr} 24 & 25 \\ 25 & 24 \end{array}\right), \left(\begin{array}{rr} 61 & 45 \\ 60 & 31 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 0 & 8 \\ 4 & 0 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 31 & 45 \\ 60 & 61 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/75\Z)$
Direct product:	$C_5$ $\, \times\, $ $D_4$ $\, \times\, $ $(C_5:C_8)$
Semidirect product:	$(C_5\times D_4)$ $\,\rtimes\,$ $C_{40}$ (4)	$C_5$ $\,\rtimes\,$ $(D_4\times C_{40})$	$(D_4\times C_5^2)$ $\,\rtimes\,$ $C_8$ (4)	$(C_{20}.D_4)$ $\,\rtimes\,$ $C_{10}$ (2)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}^2$ . $C_2^2$	$(D_4\times C_{20})$ . $D_5$	$C_{20}$ . $(D_4\times D_5)$	$C_{20}$ . $(D_4:D_5)$	all 62

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

Homology

Abelianization:	$C_{2}^{2} \times C_{40} \simeq C_{2}^{2} \times C_{8} \times C_{5}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 820 subgroups in 362 conjugacy classes, 154 normal (66 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{20}$	$G/Z \simeq$ $C_2\times D_{10}$
Commutator:	$G' \simeq$ $C_{10}$	$G/G' \simeq$ $C_2^2\times C_{40}$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_{10}\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{20}^2:C_2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{20}^2.C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1600: $C_{20}^2.C_2^2$

Order 800: $C_{10}^2:C_8$ x 2, $C_{10}^2:C_8$ x 2, $C_{20}^2:C_2$, $C_{20}:C_{40}$, $C_{20}:C_{40}$

Order 400: $C_{10}:C_{40}$ x 8, $C_2\times C_{10}\times C_{20}$ x 2, $C_{10}^2:C_4$ x 2, $C_4:C_{10}^2$, $C_{20}:C_{20}$, $C_{20}^2$

Order 320: $D_4\times C_{40}$, $C_4^2.D_{10}$

Order 200: $C_{10}\times C_{20}$ x 9, $C_5:C_{40}$ x 5, $D_4\times C_5^2$ x 4, $C_2\times C_{10}^2$ x 2

Order 160: $D_4\times C_{20}$ x 4, $C_2^2:C_{40}$ x 2, $C_{20}.D_4$ x 2, $C_2^2\times C_{40}$ x 2, $C_{20}.C_2^3$ x 2, $C_{20}:C_8$, $C_4:C_{40}$, $C_4\times C_{40}$, $C_{20}:C_8$

Order 100: $C_{10}^2$ x 9, $C_5\times C_{20}$ x 7

Order 80: $C_{10}:C_8$ x 8, $C_2^2\times C_{20}$ x 8, $C_2\times C_{40}$ x 8, $C_2^2:C_{20}$ x 8, $D_4\times C_{10}$ x 4, $C_4:C_{20}$ x 4, $C_4\times C_{20}$ x 4

Order 64: $C_8\times D_4$

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

Order 40: $C_2\times C_{20}$ x 36, $C_5\times D_4$ x 16, $C_2^2\times C_{10}$ x 8, $C_{40}$ x 5, $C_5:C_8$ x 5

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 36, $C_{20}$ x 28

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 10: $C_{10}$ x 28

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $C_{20}^2.C_2^2$

Order 800: $C_{20}^2:C_2$, $C_{10}^2:C_8$, $C_{10}^2:C_8$, $C_{20}:C_{40}$, $C_{20}:C_{40}$

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

Order 320: $D_4\times C_{40}$, $C_4^2.D_{10}$

Order 200: $C_{10}\times C_{20}$ x 6, $C_5:C_{40}$ x 3, $C_2\times C_{10}^2$, $D_4\times C_5^2$

Order 160: $D_4\times C_{20}$ x 3, $C_{20}:C_8$, $C_4:C_{40}$, $C_2^2:C_{40}$, $C_4\times C_{40}$, $C_{20}.D_4$, $C_2^2\times C_{40}$, $C_{20}.C_2^3$, $C_{20}:C_8$

Order 100: $C_5\times C_{20}$ x 5, $C_{10}^2$ x 4

Order 80: $C_{10}:C_8$ x 4, $C_2\times C_{40}$ x 4, $D_4\times C_{10}$ x 3, $C_2^2\times C_{20}$ x 3, $C_4:C_{20}$ x 3, $C_2^2:C_{20}$ x 3, $C_4\times C_{20}$ x 3

Order 64: $C_8\times D_4$

Order 50: $C_5\times C_{10}$ x 4

Order 40: $C_2\times C_{20}$ x 18, $C_{40}$ x 3, $C_2^2\times C_{10}$ x 3, $C_5\times D_4$ x 3, $C_5:C_8$ x 3

Order 32: $C_2^2:C_8$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 15, $C_2\times C_{10}$ x 12

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 10: $C_{10}$ x 12

Order 8: $C_2\times C_4$ x 6, $C_8$ x 3, $C_2^3$, $D_4$

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $C_{20}^2.C_2^2$ ($C_1$)

Order 800: $C_{10}^2:C_8$ ($C_2$) x 2, $C_{10}^2:C_8$ ($C_2$) x 2, $C_{20}^2:C_2$ ($C_2$), $C_{20}:C_{40}$ ($C_2$), $C_{20}:C_{40}$ ($C_2$)

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

Order 320: $C_4^2.D_{10}$ ($C_5$)

Order 200: $D_4\times C_5^2$ ($C_8$) x 4, $C_{10}\times C_{20}$ ($C_2\times C_4$) x 4, $C_2\times C_{10}^2$ ($C_2\times C_4$) x 2, $C_5:C_{40}$ ($D_4$) x 2, $C_{10}\times C_{20}$ ($C_2^3$)

Order 160: $C_{20}.D_4$ ($C_{10}$) x 2, $C_{20}.C_2^3$ ($C_{10}$) x 2, $C_{20}:C_8$ ($C_{10}$), $D_4\times C_{20}$ ($C_{10}$), $D_4\times C_{20}$ ($D_5$), $C_{20}:C_8$ ($C_{10}$)

Order 100: $C_{10}^2$ ($C_2\times C_8$) x 4, $C_5\times C_{20}$ ($C_2\times C_8$) x 2, $C_5\times C_{20}$ ($D_4:C_2$), $C_5\times C_{20}$ ($C_2\times D_4$), $C_{10}^2$ ($C_2^2\times C_4$)

Order 80: $C_{10}:C_8$ ($C_2\times C_{10}$) x 4, $C_2^2\times C_{20}$ ($C_2\times C_{10}$) x 2, $C_2^2\times C_{20}$ ($D_{10}$) x 2, $C_2^2:C_{20}$ ($C_{20}$) x 2, $C_2^2:C_{20}$ ($C_5:C_4$) x 2, $D_4\times C_{10}$ ($C_{20}$), $D_4\times C_{10}$ ($C_5:C_4$), $C_4:C_{20}$ ($C_{20}$), $C_4:C_{20}$ ($C_5:C_4$), $C_4\times C_{20}$ ($C_2\times C_{10}$), $C_4\times C_{20}$ ($D_{10}$)

Order 50: $C_5\times C_{10}$ ($\OD_{16}:C_2$), $C_5\times C_{10}$ ($C_2^2\times C_8$), $C_5\times C_{10}$ ($C_4\times D_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_{20}$) x 4, $C_2\times C_{20}$ ($C_{10}:C_4$) x 4, $C_5\times D_4$ ($C_{40}$) x 4, $C_5\times D_4$ ($C_5:C_8$) x 4, $C_2^2\times C_{10}$ ($C_2\times C_{20}$) x 2, $C_2^2\times C_{10}$ ($C_{10}:C_4$) x 2, $C_5:C_8$ ($C_5\times D_4$) x 2, $C_2\times C_{20}$ ($C_2^2\times C_{10}$), $C_2\times C_{20}$ ($C_2\times D_{10}$)

Order 32: $C_4\times D_4$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_8\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_{10}:C_8$) x 4, $C_2\times C_{10}$ ($C_2\times C_{40}$) x 4, $C_{20}$ ($C_{10}:C_8$) x 2, $C_{20}$ ($C_2\times C_{40}$) x 2, $C_2\times C_{10}$ ($C_2^2\times C_{20}$), $C_2\times C_{10}$ ($C_2^2.D_{10}$), $C_{20}$ ($D_4:C_{10}$), $C_{20}$ ($D_4\times C_{10}$), $C_{20}$ ($D_4:D_5$), $C_{20}$ ($D_4\times D_5$)

Order 16: $C_2^2:C_4$ ($C_5:C_{20}$) x 2, $C_2^2\times C_4$ ($C_5\times D_{10}$) x 2, $C_4:C_4$ ($C_5:C_{20}$), $C_4^2$ ($C_5\times D_{10}$), $C_2\times D_4$ ($C_5:C_{20}$)

Order 10: $C_{10}$ ($\OD_{16}:C_{10}$), $C_{10}$ ($C_2^2\times C_{40}$), $C_{10}$ ($D_4\times C_{20}$), $C_{10}$ ($C_{20}.C_2^3$), $C_{10}$ ($C_2^3.D_{10}$), $C_{10}$ ($C_{20}.C_2^3$)

Order 8: $D_4$ ($C_5:C_{40}$) x 4, $C_2\times C_4$ ($C_{10}:C_{20}$) x 4, $C_2^3$ ($C_{10}:C_{20}$) x 2, $C_2\times C_4$ ($C_{10}\times D_{10}$)

Order 5: $C_5$ ($D_4\times C_{40}$), $C_5$ ($C_4^2.D_{10}$)

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

Order 2: $C_2$ ($C_5\times C_{20}.C_2^3$), $C_2$ ($C_{10}^2.C_2^3$), $C_2$ ($C_{10}^2:C_8$)

Order 1: $C_1$ ($C_{20}^2.C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $C_{20}^2.C_2^2$ ($C_1$)

Order 800: $C_{20}^2:C_2$ ($C_2$), $C_{10}^2:C_8$ ($C_2$), $C_{10}^2:C_8$ ($C_2$), $C_{20}:C_{40}$ ($C_2$), $C_{20}:C_{40}$ ($C_2$)

Order 400: $C_{10}:C_{40}$ ($C_2^2$) x 3, $C_4:C_{10}^2$ ($C_4$), $C_2\times C_{10}\times C_{20}$ ($C_2^2$), $C_{20}:C_{20}$ ($C_4$), $C_{10}^2:C_4$ ($C_4$), $C_{20}^2$ ($C_2^2$)

Order 320: $C_4^2.D_{10}$ ($C_5$)

Order 200: $C_{10}\times C_{20}$ ($C_2\times C_4$) x 3, $C_2\times C_{10}^2$ ($C_2\times C_4$), $D_4\times C_5^2$ ($C_8$), $C_{10}\times C_{20}$ ($C_2^3$), $C_5:C_{40}$ ($D_4$)

Order 160: $C_{20}:C_8$ ($C_{10}$), $C_{20}.D_4$ ($C_{10}$), $D_4\times C_{20}$ ($C_{10}$), $D_4\times C_{20}$ ($D_5$), $C_{20}.C_2^3$ ($C_{10}$), $C_{20}:C_8$ ($C_{10}$)

Order 100: $C_5\times C_{20}$ ($C_2\times C_8$), $C_5\times C_{20}$ ($D_4:C_2$), $C_5\times C_{20}$ ($C_2\times D_4$), $C_{10}^2$ ($C_2\times C_8$), $C_{10}^2$ ($C_2^2\times C_4$)

Order 80: $C_{10}:C_8$ ($C_2\times C_{10}$) x 3, $D_4\times C_{10}$ ($C_{20}$), $D_4\times C_{10}$ ($C_5:C_4$), $C_2^2\times C_{20}$ ($C_2\times C_{10}$), $C_2^2\times C_{20}$ ($D_{10}$), $C_4:C_{20}$ ($C_{20}$), $C_4:C_{20}$ ($C_5:C_4$), $C_2^2:C_{20}$ ($C_{20}$), $C_2^2:C_{20}$ ($C_5:C_4$), $C_4\times C_{20}$ ($C_2\times C_{10}$), $C_4\times C_{20}$ ($D_{10}$)

Order 50: $C_5\times C_{10}$ ($\OD_{16}:C_2$), $C_5\times C_{10}$ ($C_2^2\times C_8$), $C_5\times C_{10}$ ($C_4\times D_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_{20}$) x 3, $C_2\times C_{20}$ ($C_{10}:C_4$) x 3, $C_2\times C_{20}$ ($C_2^2\times C_{10}$), $C_2\times C_{20}$ ($C_2\times D_{10}$), $C_2^2\times C_{10}$ ($C_2\times C_{20}$), $C_2^2\times C_{10}$ ($C_{10}:C_4$), $C_5\times D_4$ ($C_{40}$), $C_5\times D_4$ ($C_5:C_8$), $C_5:C_8$ ($C_5\times D_4$)

Order 32: $C_4\times D_4$ ($C_5\times D_5$)

Order 25: $C_5^2$ ($C_8\times D_4$)

Order 20: $C_2\times C_{10}$ ($C_{10}:C_8$), $C_2\times C_{10}$ ($C_2^2\times C_{20}$), $C_2\times C_{10}$ ($C_2^2.D_{10}$), $C_2\times C_{10}$ ($C_2\times C_{40}$), $C_{20}$ ($C_{10}:C_8$), $C_{20}$ ($D_4:C_{10}$), $C_{20}$ ($D_4\times C_{10}$), $C_{20}$ ($D_4:D_5$), $C_{20}$ ($D_4\times D_5$), $C_{20}$ ($C_2\times C_{40}$)

Order 16: $C_4:C_4$ ($C_5:C_{20}$), $C_2^2:C_4$ ($C_5:C_{20}$), $C_4^2$ ($C_5\times D_{10}$), $C_2\times D_4$ ($C_5:C_{20}$), $C_2^2\times C_4$ ($C_5\times D_{10}$)

Order 10: $C_{10}$ ($\OD_{16}:C_{10}$), $C_{10}$ ($C_2^2\times C_{40}$), $C_{10}$ ($D_4\times C_{20}$), $C_{10}$ ($C_{20}.C_2^3$), $C_{10}$ ($C_2^3.D_{10}$), $C_{10}$ ($C_{20}.C_2^3$)

Order 8: $C_2\times C_4$ ($C_{10}:C_{20}$) x 3, $C_2^3$ ($C_{10}:C_{20}$), $D_4$ ($C_5:C_{40}$), $C_2\times C_4$ ($C_{10}\times D_{10}$)

Order 5: $C_5$ ($D_4\times C_{40}$), $C_5$ ($C_4^2.D_{10}$)

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

Order 2: $C_2$ ($C_5\times C_{20}.C_2^3$), $C_2$ ($C_{10}^2.C_2^3$), $C_2$ ($C_{10}^2:C_8$)

Order 1: $C_1$ ($C_{20}^2.C_2^2$)

Series

Derived series

$C_{20}^2.C_2^2$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$C_{20}^2.C_2^2$

$\rhd$

$C_{20}:C_{40}$

$\rhd$

$C_{20}^2$

$\rhd$

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

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}^2.C_2^2$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{20}$

$\lhd$

$D_4\times C_{20}$

Character theory

Complex character table

See the $400 \times 400$ 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 $85 \times 85$ rational character table.