Abstract group 1812.13: $S_3\times C_{302}$

• Label: 1812.13
• Order: \( 2^{2} \cdot 3 \cdot 151 \)
• Exponent: \( 2 \cdot 3 \cdot 151 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 151 \)
• $\card{Z(G)}$: \( 2 \cdot 151 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{2} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 5^{2} \)
• Perm deg.: $156$
• Trans deg.: $906$
• Rank: $2$

Group information

Description:	$S_3\times C_{302}$
Order:	\(1812\)\(\medspace = 2^{2} \cdot 3 \cdot 151 \)
Exponent:	\(906\)\(\medspace = 2 \cdot 3 \cdot 151 \)
Automorphism group:	$D_6\times C_{150}$, of order \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_{151}$
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	6	151	302	453	906
Elements	1	7	2	2	150	1050	300	300	1812
Conjugacy classes	1	3	1	1	150	450	150	150	906
Divisions	1	3	1	1	1	3	1	1	12
Autjugacy classes	1	2	1	1	1	2	1	1	10

Dimension	1	2	150	300
Irr. complex chars.	604	302	0	0	906
Irr. rational chars.	4	2	4	2	12

Minimal presentations

Permutation degree:	$156$
Transitive degree:	$906$
Rank:	$2$
Inequivalent generating pairs:	$456$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	300
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{906}=1, b^{a}=b^{605} \rangle$
Permutation group:	Degree $156$ $\langle(153,154), (155,156), (1,2,4,6,8,10,12,14,16,22,20,18,19,21,23,24,25,26,27,28,33,31,29,30,32,34,35,36,37,38,39,44,42,40,41,43,45,46,47,48,49,50,55,53,51,52,54,56,57,58,59,60,61,66,64,62,63,65,67,68,69,70,71,72,77,75,73,74,76,78,79,80,81,82,83,88,86,84,85,87,89,90,91,92,93,94,99,97,95,96,98,100,101,102,103,104,105,116,114,106,111,109,107,108,110,112,113,115,117,118,119,124,122,120,121,123,125,126,127,128,129,130,135,133,131,132,134,136,137,138,139,140,141,146,144,142,143,145,147,148,149,150,151,17,15,13,11,9,7,5,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 385 & 0 \\ 0 & 523 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{907})$
Direct product:	$C_2$ $\, \times\, $ $C_{151}$ $\, \times\, $ $S_3$
Semidirect product:	$C_{906}$ $\,\rtimes\,$ $C_2$	$C_6$ $\,\rtimes\,$ $C_{302}$	$C_{453}$ $\,\rtimes\,$ $C_2^2$	$C_3$ $\,\rtimes\,$ $(C_2\times C_{302})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2} \times C_{302} \simeq C_{2}^{2} \times C_{151}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 32 subgroups in 20 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{302}$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_2\times C_{302}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times C_{302}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{906}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times C_{302}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{906}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
151-Sylow subgroup:	$P_{ 151 } \simeq$ $C_{151}$

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 1812: $S_3\times C_{302}$

Order 906: $S_3\times C_{151}$ x 2, $C_{906}$

Order 604: $C_2\times C_{302}$

Order 453: $C_{453}$

Order 302: $C_{302}$ x 3

Order 151: $C_{151}$

Order 12: $D_6$

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

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 1812: $S_3\times C_{302}$

Order 906: $C_{906}$, $S_3\times C_{151}$

Order 604: $C_2\times C_{302}$

Order 453: $C_{453}$

Order 302: $C_{302}$ x 2

Order 151: $C_{151}$

Order 12: $D_6$

Order 6: $C_6$, $S_3$

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 1812: $S_3\times C_{302}$ ($C_1$)

Order 906: $S_3\times C_{151}$ ($C_2$) x 2, $C_{906}$ ($C_2$)

Order 453: $C_{453}$ ($C_2^2$)

Order 302: $C_{302}$ ($S_3$)

Order 151: $C_{151}$ ($D_6$)

Order 12: $D_6$ ($C_{151}$)

Order 6: $S_3$ ($C_{302}$) x 2, $C_6$ ($C_{302}$)

Order 3: $C_3$ ($C_2\times C_{302}$)

Order 2: $C_2$ ($S_3\times C_{151}$)

Order 1: $C_1$ ($S_3\times C_{302}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1812: $S_3\times C_{302}$ ($C_1$)

Order 906: $C_{906}$ ($C_2$), $S_3\times C_{151}$ ($C_2$)

Order 453: $C_{453}$ ($C_2^2$)

Order 302: $C_{302}$ ($S_3$)

Order 151: $C_{151}$ ($D_6$)

Order 12: $D_6$ ($C_{151}$)

Order 6: $C_6$ ($C_{302}$), $S_3$ ($C_{302}$)

Order 3: $C_3$ ($C_2\times C_{302}$)

Order 2: $C_2$ ($S_3\times C_{151}$)

Order 1: $C_1$ ($S_3\times C_{302}$)

Series

Derived series	$S_3\times C_{302}$	$\rhd$	$C_3$	$\rhd$	$C_1$
Chief series	$S_3\times C_{302}$	$\rhd$	$S_3\times C_{151}$	$\rhd$	$C_{453}$	$\rhd$	$C_{151}$	$\rhd$	$C_1$
Lower central series	$S_3\times C_{302}$	$\rhd$	$C_3$
Upper central series	$C_1$	$\lhd$	$C_{302}$

Character theory

Complex character table

The $906 \times 906$ character table is not available for this group.

Rational character table

		1A	2A	2B	2C	3A	6A	151A	302A	302B	302C	453A	906A
	Size	1	1	3	3	2	2	150	150	450	450	300	300
	2 P	1A	1A	1A	1A	3A	3A	151A	151A	151A	151A	453A	453A
	3 P	1A	2A	2B	2C	1A	2A	151A	302A	302B	302C	151A	302A
	151 P	1A	2A	2B	2C	3A	6A	151A	302A	302B	302C	453A	906A
1812.13.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1812.13.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
1812.13.1c		$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
1812.13.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
1812.13.1e		$150$	$150$	$150$	$150$	$150$	$150$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
1812.13.1f		$150$	$-150$	$-150$	$150$	$150$	$-150$	$-1$	$1$	$-1$	$1$	$-1$	$1$
1812.13.1g		$150$	$-150$	$150$	$-150$	$150$	$-150$	$-1$	$1$	$1$	$-1$	$-1$	$1$
1812.13.1h		$150$	$150$	$-150$	$-150$	$150$	$150$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
1812.13.2a		$2$	$2$	$0$	$0$	$-1$	$-1$	$2$	$2$	$0$	$0$	$-1$	$-1$
1812.13.2b		$2$	$-2$	$0$	$0$	$-1$	$1$	$2$	$-2$	$0$	$0$	$-1$	$1$
1812.13.2c		$300$	$300$	$0$	$0$	$-150$	$-150$	$-2$	$-2$	$0$	$0$	$1$	$1$
1812.13.2d		$300$	$-300$	$0$	$0$	$-150$	$150$	$-2$	$2$	$0$	$0$	$1$	$-1$