Abstract group 20100.b: $C_{201}:C_{100}$

• Label: 20100.b
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 5^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 67 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• Perm deg.: $99$
• Trans deg.: $20100$
• Rank: $2$

Group information

Description:	$C_{201}:C_{100}$
Order:	\(20100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
Exponent:	\(20100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
Automorphism group:	$C_{201}.C_{330}.C_2^4$, of order \(1061280\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 67 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$ x 2, $C_{67}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	15	20	25	30	50	67	75	100	134	150	201	335	402	670	1005	1675	2010	3350	5025	10050
Elements	1	1	2	402	4	2	4	8	1608	20	8	20	66	40	8040	66	40	132	264	132	264	528	1320	528	1320	2640	2640	20100
Conjugacy classes	1	1	1	2	4	1	4	4	8	20	4	20	33	20	40	33	20	66	132	66	132	264	660	264	660	1320	1320	5100
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	27
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	27

Minimal presentations

Permutation degree:	$99$
Transitive degree:	$20100$
Rank:	$2$
Inequivalent generating pairs:	$180$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{100}=b^{201}=1, b^{a}=b^{200} \rangle$
Permutation group:	Degree $99$ $\langle(1,2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8,6,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 71 & 88 \\ 163 & 71 \end{array}\right), \left(\begin{array}{rr} 340 & 166 \\ 212 & 61 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{401})$
Direct product:	$C_{25}$ $\, \times\, $ $(C_{201}:C_4)$
Semidirect product:	$C_{5025}$ $\,\rtimes\,$ $C_4$	$C_{201}$ $\,\rtimes\,$ $C_{100}$	$C_{75}$ $\,\rtimes\,$ $(C_{67}:C_4)$	$C_{1675}$ $\,\rtimes\,$ $(C_3:C_4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{3350}$ . $S_3$	$C_{150}$ . $D_{67}$	$C_{50}$ . $D_{201}$	$C_{402}$ . $C_{50}$	all 17

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

Homology

Abelianization:	$C_{100} \simeq C_{4} \times C_{25}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 840 subgroups in 36 conjugacy classes, 27 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{50}$	$G/Z \simeq$ $D_{201}$
Commutator:	$G' \simeq$ $C_{201}$	$G/G' \simeq$ $C_{100}$
Frattini:	$\Phi \simeq$ $C_{10}$	$G/\Phi \simeq$ $C_5\times D_{201}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10050}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{201}:C_{100}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2010}$	$G/\operatorname{soc} \simeq$ $C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
67-Sylow subgroup:	$P_{ 67 } \simeq$ $C_{67}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 20100: $C_{201}:C_{100}$

Order 10050: $C_{10050}$

Order 6700: $C_{67}:C_{100}$

Order 5025: $C_{5025}$

Order 4020: $C_{1005}:C_4$

Order 3350: $C_{3350}$

Order 2010: $C_{2010}$

Order 1675: $C_{1675}$

Order 1340: $C_{335}:C_4$

Order 1005: $C_{1005}$

Order 804: $C_{201}:C_4$

Order 670: $C_{670}$

Order 402: $C_{402}$

Order 335: $C_{335}$

Order 300: $C_3:C_{100}$

Order 268: $C_{67}:C_4$

Order 201: $C_{201}$

Order 150: $C_{150}$

Order 134: $C_{134}$

Order 100: $C_{100}$

Order 75: $C_{75}$

Order 67: $C_{67}$

Order 60: $C_3:C_{20}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 12: $C_3:C_4$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 20100: $C_{201}:C_{100}$

Order 10050: $C_{10050}$

Order 6700: $C_{67}:C_{100}$

Order 5025: $C_{5025}$

Order 4020: $C_{1005}:C_4$

Order 3350: $C_{3350}$

Order 2010: $C_{2010}$

Order 1675: $C_{1675}$

Order 1340: $C_{335}:C_4$

Order 1005: $C_{1005}$

Order 804: $C_{201}:C_4$

Order 670: $C_{670}$

Order 402: $C_{402}$

Order 335: $C_{335}$

Order 300: $C_3:C_{100}$

Order 268: $C_{67}:C_4$

Order 201: $C_{201}$

Order 150: $C_{150}$

Order 134: $C_{134}$

Order 100: $C_{100}$

Order 75: $C_{75}$

Order 67: $C_{67}$

Order 60: $C_3:C_{20}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 12: $C_3:C_4$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20100: $C_{201}:C_{100}$ ($C_1$)

Order 10050: $C_{10050}$ ($C_2$)

Order 5025: $C_{5025}$ ($C_4$)

Order 4020: $C_{1005}:C_4$ ($C_5$)

Order 3350: $C_{3350}$ ($S_3$)

Order 2010: $C_{2010}$ ($C_{10}$)

Order 1675: $C_{1675}$ ($C_3:C_4$)

Order 1005: $C_{1005}$ ($C_{20}$)

Order 804: $C_{201}:C_4$ ($C_{25}$)

Order 670: $C_{670}$ ($C_5\times S_3$)

Order 402: $C_{402}$ ($C_{50}$)

Order 335: $C_{335}$ ($C_3:C_{20}$)

Order 201: $C_{201}$ ($C_{100}$)

Order 150: $C_{150}$ ($D_{67}$)

Order 134: $C_{134}$ ($S_3\times C_{25}$)

Order 75: $C_{75}$ ($C_{67}:C_4$)

Order 67: $C_{67}$ ($C_3:C_{100}$)

Order 50: $C_{50}$ ($D_{201}$)

Order 30: $C_{30}$ ($C_5\times D_{67}$)

Order 25: $C_{25}$ ($C_{201}:C_4$)

Order 15: $C_{15}$ ($C_{335}:C_4$)

Order 10: $C_{10}$ ($C_5\times D_{201}$)

Order 6: $C_6$ ($C_{25}\times D_{67}$)

Order 5: $C_5$ ($C_{1005}:C_4$)

Order 3: $C_3$ ($C_{67}:C_{100}$)

Order 2: $C_2$ ($C_{25}\times D_{201}$)

Order 1: $C_1$ ($C_{201}:C_{100}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20100: $C_{201}:C_{100}$ ($C_1$)

Order 10050: $C_{10050}$ ($C_2$)

Order 5025: $C_{5025}$ ($C_4$)

Order 4020: $C_{1005}:C_4$ ($C_5$)

Order 3350: $C_{3350}$ ($S_3$)

Order 2010: $C_{2010}$ ($C_{10}$)

Order 1675: $C_{1675}$ ($C_3:C_4$)

Order 1005: $C_{1005}$ ($C_{20}$)

Order 804: $C_{201}:C_4$ ($C_{25}$)

Order 670: $C_{670}$ ($C_5\times S_3$)

Order 402: $C_{402}$ ($C_{50}$)

Order 335: $C_{335}$ ($C_3:C_{20}$)

Order 201: $C_{201}$ ($C_{100}$)

Order 150: $C_{150}$ ($D_{67}$)

Order 134: $C_{134}$ ($S_3\times C_{25}$)

Order 75: $C_{75}$ ($C_{67}:C_4$)

Order 67: $C_{67}$ ($C_3:C_{100}$)

Order 50: $C_{50}$ ($D_{201}$)

Order 30: $C_{30}$ ($C_5\times D_{67}$)

Order 25: $C_{25}$ ($C_{201}:C_4$)

Order 15: $C_{15}$ ($C_{335}:C_4$)

Order 10: $C_{10}$ ($C_5\times D_{201}$)

Order 6: $C_6$ ($C_{25}\times D_{67}$)

Order 5: $C_5$ ($C_{1005}:C_4$)

Order 3: $C_3$ ($C_{67}:C_{100}$)

Order 2: $C_2$ ($C_{25}\times D_{201}$)

Order 1: $C_1$ ($C_{201}:C_{100}$)

Series

Derived series

$C_{201}:C_{100}$

$\rhd$

$C_{201}$

$\rhd$

$C_1$

Chief series

$C_{201}:C_{100}$

$\rhd$

$C_{10050}$

$\rhd$

$C_{5025}$

$\rhd$

$C_{1005}$

$\rhd$

$C_{201}$

$\rhd$

$C_{67}$

$\rhd$

$C_1$

Lower central series

$C_{201}:C_{100}$

$\rhd$

$C_{201}$

Upper central series

$C_1$

$\lhd$

$C_{50}$

Supergroups

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

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

Character theory

Complex character table

The $5100 \times 5100$ character table is not available for this group.

Rational character table

The $27 \times 27$ rational character table is not available for this group.