Abstract group 38612.a: $F_{197}$

• Label: 38612.a
• Order: \( 2^{2} \cdot 7^{2} \cdot 197 \)
• Exponent: \( 2^{2} \cdot 7^{2} \cdot 197 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 7^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 7^{2} \cdot 197 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $197$
• Trans deg.: $197$
• Rank: $2$

Group information

Description:	$F_{197}$
Order:	\(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \)
Exponent:	\(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \)
Automorphism group:	$F_{197}$, of order \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \)
Composition factors:	$C_2$ x 2, $C_7$ x 2, $C_{197}$
Derived length:	$2$

This group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

Group statistics

Order	1	2	4	7	14	28	49	98	196	197
Elements	1	197	394	1182	1182	2364	8274	8274	16548	196	38612
Conjugacy classes	1	1	2	6	6	12	42	42	84	1	197
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	2	6	6	12	42	42	84	1	197

Dimension	1	2	6	12	42	84	196
Irr. complex chars.	196	0	0	0	0	0	1	197
Irr. rational chars.	2	1	2	1	2	1	1	10

Minimal presentations

Permutation degree:	$197$
Transitive degree:	$197$
Rank:	$2$
Inequivalent generating pairs:	$28224$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	196	196	196
Arbitrary	196	196	196

Constructions

Groups of Lie type:	$\AGL(1,197)$
Presentation:	$\langle a, b \mid a^{196}=b^{197}=1, b^{a}=b^{32} \rangle$
Permutation group:	Degree $197$ $\langle(2,4,8,16,30,56,98,156,180,124,123,175,181,159,99,158,188,178,129,131,174,119,69,68,111,64,66,65,106,63,110,167,173,144,170,169,162,182,126,127,183,197,184,128,75,42,44,55,46,83,142,186,133,134,185,192,143,93,95,154,141,82,140,152,94,52,54,85,47,84,91,89,112,104,58,103,153,155,132,77,43,23,15,13,26,49,39,21,19,17,33,62,108,92,51,27,14,7,3,6,12,24,45,80,136,177,176,150,151,179,164,190,137,189,157,194,147,146,172,116,96,97,145,88,78,79,135,87,118,171,166,109,165,187,191,195,149,148,193,196,160,100,90,50,36,35,32,61,107,163,114,113,168,161,105,72,70,117,102,57,101,122,71,40,38,60,31,59,74,76,130,139,81,138,121,120,115,67,37,20,10,9,18,34,53,28,29,25,48,86,125,73,41,22,11,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 118 & 0 \\ 0 & 148 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 104 & 0 \\ 0 & 36 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{197})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{197}$ $\,\rtimes\,$ $C_{196}$	$(C_{197}:C_{49})$ $\,\rtimes\,$ $C_4$	$(C_{197}:C_4)$ $\,\rtimes\,$ $C_{49}$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{197}$ . $C_{98}$	$(C_{197}:C_{98})$ . $C_2$	$(C_{197}:C_{28})$ . $C_7$	$(C_{197}:C_7)$ . $C_{28}$	all 5
Aut. group:	$\Aut(D_{197})$	$\Aut(C_{197}:C_4)$	$\Aut(C_{197}:C_7)$	$\Aut(C_{197}:C_{14})$	all 9

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

Homology

Abelianization:	$C_{196} \simeq C_{4} \times C_{49}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1586 subgroups in 18 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_{197}$
Commutator:	$G' \simeq$ $C_{197}$	$G/G' \simeq$ $C_{196}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_{197}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{197}$	$G/\operatorname{Fit} \simeq$ $C_{196}$
Radical:	$R \simeq$ $F_{197}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{197}$	$G/\operatorname{soc} \simeq$ $C_{196}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_{49}$
197-Sylow subgroup:	$P_{ 197 } \simeq$ $C_{197}$

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 38612: $F_{197}$

Order 19306: $C_{197}:C_{98}$

Order 9653: $C_{197}:C_{49}$

Order 5516: $C_{197}:C_{28}$

Order 2758: $C_{197}:C_{14}$

Order 1379: $C_{197}:C_7$

Order 788: $C_{197}:C_4$

Order 394: $D_{197}$

Order 197: $C_{197}$

Order 196: $C_{196}$

Order 98: $C_{98}$

Order 49: $C_{49}$

Order 28: $C_{28}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 38612: $F_{197}$

Order 19306: $C_{197}:C_{98}$

Order 9653: $C_{197}:C_{49}$

Order 5516: $C_{197}:C_{28}$

Order 2758: $C_{197}:C_{14}$

Order 1379: $C_{197}:C_7$

Order 788: $C_{197}:C_4$

Order 394: $D_{197}$

Order 197: $C_{197}$

Order 196: $C_{196}$

Order 98: $C_{98}$

Order 49: $C_{49}$

Order 28: $C_{28}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 38612: $F_{197}$ ($C_1$)

Order 19306: $C_{197}:C_{98}$ ($C_2$)

Order 9653: $C_{197}:C_{49}$ ($C_4$)

Order 5516: $C_{197}:C_{28}$ ($C_7$)

Order 2758: $C_{197}:C_{14}$ ($C_{14}$)

Order 1379: $C_{197}:C_7$ ($C_{28}$)

Order 788: $C_{197}:C_4$ ($C_{49}$)

Order 394: $D_{197}$ ($C_{98}$)

Order 197: $C_{197}$ ($C_{196}$)

Order 1: $C_1$ ($F_{197}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 38612: $F_{197}$ ($C_1$)

Order 19306: $C_{197}:C_{98}$ ($C_2$)

Order 9653: $C_{197}:C_{49}$ ($C_4$)

Order 5516: $C_{197}:C_{28}$ ($C_7$)

Order 2758: $C_{197}:C_{14}$ ($C_{14}$)

Order 1379: $C_{197}:C_7$ ($C_{28}$)

Order 788: $C_{197}:C_4$ ($C_{49}$)

Order 394: $D_{197}$ ($C_{98}$)

Order 197: $C_{197}$ ($C_{196}$)

Order 1: $C_1$ ($F_{197}$)

Series

Derived series

$F_{197}$

$\rhd$

$C_{197}$

$\rhd$

$C_1$

Chief series

$F_{197}$

$\rhd$

$C_{197}:C_{98}$

$\rhd$

$C_{197}:C_{49}$

$\rhd$

$C_{197}:C_7$

$\rhd$

$C_{197}$

$\rhd$

$C_1$

Lower central series

$F_{197}$

$\rhd$

$C_{197}$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $197 \times 197$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	7A	14A	28A	49A	98A	196A	197A
	Size	1	197	394	1182	1182	2364	8274	8274	16548	196
	2 P	1A	1A	2A	7A	7A	14A	49A	49A	98A	197A
	7 P	1A	2A	4A	7A	14A	28A	49A	98A	196A	197A
	197 P	1A	2A	4A	1A	2A	4A	7A	14A	28A	197A
38612.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
38612.a.1b		$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$
38612.a.1c		$2$	$-2$	$0$	$2$	$-2$	$0$	$2$	$-2$	$0$	$2$
38612.a.1d		$6$	$6$	$6$	$6$	$6$	$6$	$-1$	$-1$	$-1$	$6$
38612.a.1e		$6$	$6$	$-6$	$6$	$6$	$-6$	$-1$	$-1$	$1$	$6$
38612.a.1f		$12$	$-12$	$0$	$12$	$-12$	$0$	$-2$	$2$	$0$	$12$
38612.a.1g		$42$	$42$	$42$	$-7$	$-7$	$-7$	$0$	$0$	$0$	$42$
38612.a.1h		$42$	$42$	$-42$	$-7$	$-7$	$7$	$0$	$0$	$0$	$42$
38612.a.1i		$84$	$-84$	$0$	$-14$	$14$	$0$	$0$	$0$	$0$	$84$
38612.a.196a		$196$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$