Abstract group 20100.a: $C_{25}\times D_{402}$

• Label: 20100.a
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
• Exponent: \( 2 \cdot 3 \cdot 5^{2} \cdot 67 \)
• 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.: $97$
• Trans deg.: $10050$
• Rank: $2$

Group information

Description:	$C_{25}\times D_{402}$
Order:	\(20100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 67 \)
Exponent:	\(10050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 67 \)
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, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	3	5	6	10	15	25	30	50	67	75	134	150	201	335	402	670	1005	1675	2010	3350	5025	10050
Elements	1	403	2	4	2	1612	8	20	8	8060	66	40	66	40	132	264	132	264	528	1320	528	1320	2640	2640	20100
Conjugacy classes	1	3	1	4	1	12	4	20	4	60	33	20	33	20	66	132	66	132	264	660	264	660	1320	1320	5100
Divisions	1	3	1	1	1	3	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	1	30
Autjugacy classes	1	2	1	1	1	2	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	27

Minimal presentations

Permutation degree:	$97$
Transitive degree:	$10050$
Rank:	$2$
Inequivalent generating pairs:	$90$

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:	$\langle a, b \mid a^{2}=b^{10050}=1, b^{a}=b^{401} \rangle$
Permutation group:	Degree $97$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 400 \end{array}\right), \left(\begin{array}{rr} 71 & 88 \\ 163 & 71 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{401})$
Direct product:	$C_2$ $\, \times\, $ $C_{25}$ $\, \times\, $ $D_{201}$
Semidirect product:	$C_{3350}$ $\,\rtimes\,$ $S_3$	$C_{1675}$ $\,\rtimes\,$ $D_6$	$C_{150}$ $\,\rtimes\,$ $D_{67}$	$C_{75}$ $\,\rtimes\,$ $D_{134}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{2010}$ . $C_{10}$	$C_{670}$ . $(C_5\times S_3)$	$(C_5\times D_{402})$ . $C_5$	$C_{30}$ . $(C_5\times D_{67})$	all 10

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

Homology

Abelianization:	$C_{2} \times C_{50} \simeq C_{2}^{2} \times C_{25}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 2472 subgroups in 60 conjugacy classes, 33 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{50}$	$G/Z \simeq$ $D_{201}$
Commutator:	$G' \simeq$ $C_{201}$	$G/G' \simeq$ $C_2\times C_{50}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $C_5\times D_{402}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10050}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{25}\times D_{402}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2010}$	$G/\operatorname{soc} \simeq$ $C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 20100: $C_{25}\times D_{402}$

Order 10050: $C_{25}\times D_{201}$ x 2, $C_{10050}$

Order 6700: $C_{25}\times D_{134}$

Order 5025: $C_{5025}$

Order 4020: $C_5\times D_{402}$

Order 3350: $C_{25}\times D_{67}$ x 2, $C_{3350}$

Order 2010: $C_5\times D_{201}$ x 2, $C_{2010}$

Order 1675: $C_{1675}$

Order 1340: $C_5\times D_{134}$

Order 1005: $C_{1005}$

Order 804: $D_{402}$

Order 670: $C_5\times D_{67}$ x 2, $C_{670}$

Order 402: $D_{201}$ x 2, $C_{402}$

Order 335: $C_{335}$

Order 300: $S_3\times C_{50}$

Order 268: $D_{134}$

Order 201: $C_{201}$

Order 150: $S_3\times C_{25}$ x 2, $C_{150}$

Order 134: $D_{67}$ x 2, $C_{134}$

Order 100: $C_2\times C_{50}$

Order 75: $C_{75}$

Order 67: $C_{67}$

Order 60: $S_3\times C_{10}$

Order 50: $C_{50}$ x 3

Order 30: $C_5\times S_3$ x 2, $C_{30}$

Order 25: $C_{25}$

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

Order 15: $C_{15}$

Order 12: $D_6$

Order 10: $C_{10}$ x 3

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 20100: $C_{25}\times D_{402}$

Order 10050: $C_{10050}$, $C_{25}\times D_{201}$

Order 6700: $C_{25}\times D_{134}$

Order 5025: $C_{5025}$

Order 4020: $C_5\times D_{402}$

Order 3350: $C_{25}\times D_{67}$, $C_{3350}$

Order 2010: $C_5\times D_{201}$, $C_{2010}$

Order 1675: $C_{1675}$

Order 1340: $C_5\times D_{134}$

Order 1005: $C_{1005}$

Order 804: $D_{402}$

Order 670: $C_{670}$, $C_5\times D_{67}$

Order 402: $C_{402}$, $D_{201}$

Order 335: $C_{335}$

Order 300: $S_3\times C_{50}$

Order 268: $D_{134}$

Order 201: $C_{201}$

Order 150: $C_{150}$, $S_3\times C_{25}$

Order 134: $C_{134}$, $D_{67}$

Order 100: $C_2\times C_{50}$

Order 75: $C_{75}$

Order 67: $C_{67}$

Order 60: $S_3\times C_{10}$

Order 50: $C_{50}$ x 2

Order 30: $C_{30}$, $C_5\times S_3$

Order 25: $C_{25}$

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

Order 15: $C_{15}$

Order 12: $D_6$

Order 10: $C_{10}$ x 2

Order 6: $C_6$, $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20100: $C_{25}\times D_{402}$ ($C_1$)

Order 10050: $C_{25}\times D_{201}$ ($C_2$) x 2, $C_{10050}$ ($C_2$)

Order 5025: $C_{5025}$ ($C_2^2$)

Order 4020: $C_5\times D_{402}$ ($C_5$)

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

Order 2010: $C_5\times D_{201}$ ($C_{10}$) x 2, $C_{2010}$ ($C_{10}$)

Order 1675: $C_{1675}$ ($D_6$)

Order 1005: $C_{1005}$ ($C_2\times C_{10}$)

Order 804: $D_{402}$ ($C_{25}$)

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

Order 402: $D_{201}$ ($C_{50}$) x 2, $C_{402}$ ($C_{50}$)

Order 335: $C_{335}$ ($S_3\times C_{10}$)

Order 201: $C_{201}$ ($C_2\times C_{50}$)

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

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

Order 75: $C_{75}$ ($D_{134}$)

Order 67: $C_{67}$ ($S_3\times C_{50}$)

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

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

Order 25: $C_{25}$ ($D_{402}$)

Order 15: $C_{15}$ ($C_5\times D_{134}$)

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

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

Order 5: $C_5$ ($C_5\times D_{402}$)

Order 3: $C_3$ ($C_{25}\times D_{134}$)

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

Order 1: $C_1$ ($C_{25}\times D_{402}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20100: $C_{25}\times D_{402}$ ($C_1$)

Order 10050: $C_{10050}$ ($C_2$), $C_{25}\times D_{201}$ ($C_2$)

Order 5025: $C_{5025}$ ($C_2^2$)

Order 4020: $C_5\times D_{402}$ ($C_5$)

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

Order 2010: $C_5\times D_{201}$ ($C_{10}$), $C_{2010}$ ($C_{10}$)

Order 1675: $C_{1675}$ ($D_6$)

Order 1005: $C_{1005}$ ($C_2\times C_{10}$)

Order 804: $D_{402}$ ($C_{25}$)

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

Order 402: $C_{402}$ ($C_{50}$), $D_{201}$ ($C_{50}$)

Order 335: $C_{335}$ ($S_3\times C_{10}$)

Order 201: $C_{201}$ ($C_2\times C_{50}$)

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

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

Order 75: $C_{75}$ ($D_{134}$)

Order 67: $C_{67}$ ($S_3\times C_{50}$)

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

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

Order 25: $C_{25}$ ($D_{402}$)

Order 15: $C_{15}$ ($C_5\times D_{134}$)

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

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

Order 5: $C_5$ ($C_5\times D_{402}$)

Order 3: $C_3$ ($C_{25}\times D_{134}$)

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

Order 1: $C_1$ ($C_{25}\times D_{402}$)

Series

Derived series

$C_{25}\times D_{402}$

$\rhd$

$C_{201}$

$\rhd$

$C_1$

Chief series

$C_{25}\times D_{402}$

$\rhd$

$C_{25}\times D_{201}$

$\rhd$

$C_{5025}$

$\rhd$

$C_{1675}$

$\rhd$

$C_{335}$

$\rhd$

$C_{67}$

$\rhd$

$C_1$

Lower central series

$C_{25}\times D_{402}$

$\rhd$

$C_{201}$

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 $30 \times 30$ rational character table is not available for this group.