Abstract group 1600.299: $(C_2\times C_4).D_{100}$

• Label: 1600.299
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 5 \)
• Perm deg.: $37$
• Trans deg.: $800$
• Rank: $3$

Group information

Description:	$(C_2\times C_4).D_{100}$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Automorphism group:	Group of order \(256000\)\(\medspace = 2^{11} \cdot 5^{3} \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20	25	50	100
Elements	1	207	624	4	28	96	20	140	480	1600
Conjugacy classes	1	9	18	2	14	24	10	70	120	268
Divisions	1	9	10	1	7	5	1	7	5	46
Autjugacy classes	1	8	6	1	7	3	1	7	3	37

Dimension	1	2	4	8	16	20	40	80
Irr. complex chars.	16	204	48	0	0	0	0	0	268
Irr. rational chars.	8	8	8	8	1	4	8	1	46

Minimal presentations

Permutation degree:	$37$
Transitive degree:	$800$
Rank:	$3$
Inequivalent generating triples:	$5040$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	6	26

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{2}=d^{100}=[a,c]=[b,c]=[c,d]=1, b^{a}=bcd^{50}, d^{a}=cd^{49}, d^{b}=cd^{51} \rangle$
Permutation group:	Degree $37$ $\langle(2,4)(5,6)(7,8)(9,10)(11,12)(14,16)(15,19)(17,22)(18,25)(20,28)(21,27)(23,30) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_{50}:C_4)$ $\,\rtimes\,$ $C_4$ (4)	$(C_2.C_4^2)$ $\,\rtimes\,$ $D_{25}$	$(C_{50}.C_4^2)$ $\,\rtimes\,$ $C_2$	$(C_{50}:C_4^2)$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{50}:C_4)$ . $D_4$ (2)	$C_{50}$ . $(C_4\times D_4)$ (2)	$C_{50}$ . $(C_4:D_4)$	$(C_2\times C_{100})$ . $D_4$ (2)	all 55

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

Homology

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

Subgroups

There are 3591 subgroups in 285 conjugacy classes, 93 normal (71 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2\times D_{50}$
Commutator:	$G' \simeq$ $C_2\times C_{50}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_{10}$	$G/\Phi \simeq$ $C_2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{50}.C_4^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_4).D_{100}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times D_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$

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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 1600: $(C_2\times C_4).D_{100}$

Order 800: $C_2^2.D_{100}$ x 3, $C_{50}.C_4^2$, $C_{50}.C_4^2$, $C_2^2.D_{100}$, $C_{50}:C_4^2$

Order 400: $D_{50}:C_4$ x 8, $C_2^2\times C_{100}$ x 3, $C_2^2.D_{50}$ x 3, $C_{100}:C_4$ x 2, $C_{100}:C_4$ x 2, $C_2^2\times D_{50}$

Order 320: $(C_2\times C_4).D_{20}$

Order 200: $C_2\times C_{100}$ x 11, $C_{50}:C_4$ x 11, $C_2\times D_{50}$ x 8, $C_2^2\times C_{50}$

Order 160: $C_2^2.D_{20}$ x 3, $C_{10}.C_4^2$, $C_{10}.C_4^2$, $C_2^2.D_{20}$, $C_{10}:C_4^2$

Order 100: $D_{50}$ x 10, $C_2\times C_{50}$ x 7, $C_{100}$ x 5, $C_{25}:C_4$ x 5

Order 80: $D_{10}:C_4$ x 8, $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3, $C_{20}:C_4$ x 2, $C_{20}:C_4$ x 2, $C_2^2\times D_{10}$

Order 64: $C_2^3.C_2^3$

Order 50: $C_{50}$ x 7, $D_{25}$ x 2

Order 40: $C_2\times C_{20}$ x 11, $C_{10}:C_4$ x 11, $C_2\times D_{10}$ x 8, $C_2^2\times C_{10}$

Order 32: $C_2^3:C_4$ x 3, $C_2.C_4^2$ x 2, $C_2^2.D_4$, $C_2\times C_4^2$

Order 25: $C_{25}$

Order 20: $D_{10}$ x 10, $C_2\times C_{10}$ x 7, $C_{20}$ x 5, $C_5:C_4$ x 5

Order 16: $C_2^2:C_4$ x 8, $C_2^2\times C_4$ x 6, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2^4$

Order 10: $C_{10}$ x 7, $D_5$ x 2

Order 8: $C_2\times C_4$ x 22, $C_2^3$ x 9

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 10

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $(C_2\times C_4).D_{100}$

Order 800: $C_2^2.D_{100}$ x 3, $C_{50}.C_4^2$, $C_{50}.C_4^2$, $C_2^2.D_{100}$, $C_{50}:C_4^2$

Order 400: $C_2^2\times C_{100}$ x 3, $C_2^2.D_{50}$ x 3, $D_{50}:C_4$ x 3, $C_2^2\times D_{50}$, $C_{100}:C_4$, $C_{100}:C_4$

Order 320: $(C_2\times C_4).D_{20}$

Order 200: $C_2\times C_{100}$ x 9, $C_{50}:C_4$ x 9, $C_2\times D_{50}$ x 7, $C_2^2\times C_{50}$

Order 160: $C_2^2.D_{20}$ x 3, $C_{10}.C_4^2$, $C_{10}.C_4^2$, $C_2^2.D_{20}$, $C_{10}:C_4^2$

Order 100: $C_2\times C_{50}$ x 7, $D_{50}$ x 7, $C_{100}$ x 3, $C_{25}:C_4$ x 3

Order 80: $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3, $D_{10}:C_4$ x 3, $C_2^2\times D_{10}$, $C_{20}:C_4$, $C_{20}:C_4$

Order 64: $C_2^3.C_2^3$

Order 50: $C_{50}$ x 7, $D_{25}$

Order 40: $C_2\times C_{20}$ x 9, $C_{10}:C_4$ x 9, $C_2\times D_{10}$ x 7, $C_2^2\times C_{10}$

Order 32: $C_2^3:C_4$ x 3, $C_2.C_4^2$ x 2, $C_2^2.D_4$, $C_2\times C_4^2$

Order 25: $C_{25}$

Order 20: $C_2\times C_{10}$ x 7, $D_{10}$ x 7, $C_{20}$ x 3, $C_5:C_4$ x 3

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

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

Order 8: $C_2\times C_4$ x 18, $C_2^3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 14, $C_4$ x 6

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $(C_2\times C_4).D_{100}$ ($C_1$)

Order 800: $C_2^2.D_{100}$ ($C_2$) x 3, $C_{50}.C_4^2$ ($C_2$), $C_{50}.C_4^2$ ($C_2$), $C_2^2.D_{100}$ ($C_2$), $C_{50}:C_4^2$ ($C_2$)

Order 400: $D_{50}:C_4$ ($C_4$) x 4, $C_2^2\times C_{100}$ ($C_2^2$) x 3, $C_2^2.D_{50}$ ($C_2^2$) x 3, $C_2^2\times D_{50}$ ($C_2^2$)

Order 200: $C_2\times C_{100}$ ($D_4$) x 2, $C_2\times C_{100}$ ($C_2\times C_4$) x 2, $C_{50}:C_4$ ($D_4$) x 2, $C_{50}:C_4$ ($C_2\times C_4$) x 2, $C_2\times D_{50}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{50}$ ($C_2^3$)

Order 160: $C_{10}.C_4^2$ ($D_5$)

Order 100: $C_2\times C_{50}$ ($D_4:C_2$) x 4, $C_2\times C_{50}$ ($C_2\times D_4$) x 2, $C_2\times C_{50}$ ($C_2^2\times C_4$)

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

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

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

Order 32: $C_2.C_4^2$ ($D_{25}$)

Order 25: $C_{25}$ ($C_2^3.C_2^3$)

Order 20: $C_2\times C_{10}$ ($D_4:D_5$) x 2, $C_2\times C_{10}$ ($D_{20}:C_2$), $C_2\times C_{10}$ ($D_4\times D_5$), $C_2\times C_{10}$ ($D_{20}:C_2$), $C_2\times C_{10}$ ($C_2\times D_{20}$), $C_2\times C_{10}$ ($C_4\times D_{10}$)

Order 16: $C_2^2\times C_4$ ($D_{50}$) x 3

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

Order 8: $C_2\times C_4$ ($D_{100}$) x 2, $C_2\times C_4$ ($C_4\times D_{25}$) x 2, $C_2^3$ ($C_2\times D_{50}$)

Order 5: $C_5$ ($(C_2\times C_4).D_{20}$)

Order 4: $C_2^2$ ($D_4:D_{25}$) x 2, $C_2^2$ ($D_{100}:C_2$), $C_2^2$ ($D_4\times D_{25}$), $C_2^2$ ($D_{100}:C_2$), $C_2^2$ ($C_2\times D_{100}$), $C_2^2$ ($C_4\times D_{50}$)

Order 2: $C_2$ ($C_4\times D_{100}$), $C_2$ ($(C_2\times C_4).D_{50}$), $C_2$ ($C_4:D_{100}$), $C_2$ ($(C_4\times D_{25}):C_4$), $C_2$ ($C_2^2.D_{100}$), $C_2$ ($C_2^3.D_{50}$), $C_2$ ($C_2^3.D_{50}$)

Order 1: $C_1$ ($(C_2\times C_4).D_{100}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $(C_2\times C_4).D_{100}$ ($C_1$)

Order 800: $C_2^2.D_{100}$ ($C_2$) x 3, $C_{50}.C_4^2$ ($C_2$), $C_{50}.C_4^2$ ($C_2$), $C_2^2.D_{100}$ ($C_2$), $C_{50}:C_4^2$ ($C_2$)

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

Order 200: $C_2\times C_{100}$ ($D_4$), $C_2\times C_{100}$ ($C_2\times C_4$), $C_{50}:C_4$ ($D_4$), $C_{50}:C_4$ ($C_2\times C_4$), $C_2^2\times C_{50}$ ($C_2^3$), $C_2\times D_{50}$ ($C_2\times C_4$)

Order 160: $C_{10}.C_4^2$ ($D_5$)

Order 100: $C_2\times C_{50}$ ($D_4:C_2$) x 4, $C_2\times C_{50}$ ($C_2\times D_4$) x 2, $C_2\times C_{50}$ ($C_2^2\times C_4$)

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

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

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

Order 32: $C_2.C_4^2$ ($D_{25}$)

Order 25: $C_{25}$ ($C_2^3.C_2^3$)

Order 20: $C_2\times C_{10}$ ($D_4:D_5$) x 2, $C_2\times C_{10}$ ($D_{20}:C_2$), $C_2\times C_{10}$ ($D_4\times D_5$), $C_2\times C_{10}$ ($D_{20}:C_2$), $C_2\times C_{10}$ ($C_2\times D_{20}$), $C_2\times C_{10}$ ($C_4\times D_{10}$)

Order 16: $C_2^2\times C_4$ ($D_{50}$) x 3

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

Order 8: $C_2^3$ ($C_2\times D_{50}$), $C_2\times C_4$ ($D_{100}$), $C_2\times C_4$ ($C_4\times D_{25}$)

Order 5: $C_5$ ($(C_2\times C_4).D_{20}$)

Order 4: $C_2^2$ ($D_4:D_{25}$) x 2, $C_2^2$ ($D_{100}:C_2$), $C_2^2$ ($D_4\times D_{25}$), $C_2^2$ ($D_{100}:C_2$), $C_2^2$ ($C_2\times D_{100}$), $C_2^2$ ($C_4\times D_{50}$)

Order 2: $C_2$ ($C_4\times D_{100}$), $C_2$ ($(C_2\times C_4).D_{50}$), $C_2$ ($C_4:D_{100}$), $C_2$ ($(C_4\times D_{25}):C_4$), $C_2$ ($C_2^2.D_{100}$), $C_2$ ($C_2^3.D_{50}$), $C_2$ ($C_2^3.D_{50}$)

Order 1: $C_1$ ($(C_2\times C_4).D_{100}$)

Series

Derived series

$(C_2\times C_4).D_{100}$

$\rhd$

$C_2\times C_{50}$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4).D_{100}$

$\rhd$

$C_2^2.D_{100}$

$\rhd$

$C_2^2\times D_{50}$

$\rhd$

$C_2^2\times C_{50}$

$\rhd$

$C_2\times C_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4).D_{100}$

$\rhd$

$C_2\times C_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2.C_4^2$

Character theory

Complex character table

See the $268 \times 268$ 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 $46 \times 46$ rational character table.