Abstract group 74112.g: $D_{193}:C_{192}$

• Label: 74112.g
• Order: \( 2^{7} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $260$
• Trans deg.: $37056$
• Rank: $2$

Group information

Description:	$D_{193}:C_{192}$
Order:	\(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{386}.C_{96}.C_2^4$, of order \(592896\)\(\medspace = 2^{10} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 7, $C_3$, $C_{193}$
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	4	6	8	12	16	24	32	48	64	96	192	193	386	579	772	1158	2316
Elements	1	387	2	388	774	1544	776	3088	3088	6176	6176	12352	12352	24704	192	192	384	384	384	768	74112
Conjugacy classes	1	3	2	4	6	8	8	16	16	32	32	64	64	128	6	6	12	12	12	24	456
Divisions	1	3	1	2	3	2	2	2	2	2	2	2	2	2	1	1	1	1	1	1	34
Autjugacy classes	1	2	1	3	2	6	3	12	6	16	12	16	16	16	1	1	1	1	1	1	118

Dimension	1	2	4	8	16	32	64	192	384	768
Irr. complex chars.	384	0	0	0	0	72	0	0	0	0	456
Irr. rational chars.	4	6	4	4	4	4	2	2	3	1	34

Minimal presentations

Permutation degree:	$260$
Transitive degree:	$37056$
Rank:	$2$
Inequivalent generating pairs:	$3072$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	64	768
Arbitrary	32	34	226

Constructions

Presentation:	$\langle a, b \mid b^{2316}=1, a^{32}=b^{386}, b^{a}=b^{217} \rangle$
Permutation group:	Degree $260$ $\langle(1,2,4,12,38,102,97,36,11,35,78,165,86,60,18,59,34,99,120,172,157,69,74,23,6,21,67,85,43,114,51,15,50,127,164,173,158,137,57,136,187,181,107,89,174,153,125,48,14,46,39,106,113,126,117,45,88,130,191,176,166,79,105,92,30,91,162,75,84,27,83,109,40,108,163,177,171,82,146,62,19,5,17,56,121,170,186,193,140,143,183,167,184,115,155,68,104,160,70,159,123,188,148,189,149,142,192,185,135,119,81,26,7,25,77,139,58,138,156,141,131,71,151,65,20,64,73,133,55,37,101,103,63,147,134,190,129,100,52,118,124,128,111,41,98,33,96,179,161,175,90,110,182,180,169,80,168,154,150,145,116,44,13,42,112,49,66,95,32,9,31,93,94,178,152,132,54,16,53,24,76,87,144,61,29,8,28,47,122,72,22,10,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 163 & 0 \\ 0 & 140 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 9 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_3$ $\, \times\, $ $(D_{193}:C_{64})$
Semidirect product:	$D_{193}$ $\,\rtimes\,$ $C_{192}$ (2)	$(C_3\times D_{193})$ $\,\rtimes\,$ $C_{64}$ (2)	$(C_{193}:C_{64})$ $\,\rtimes\,$ $C_6$ (2)	$C_{579}$ $\,\rtimes\,$ $(C_2\times C_{64})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{386}$ . $C_{96}$	$C_{772}$ . $C_{96}$	$C_{2316}$ . $C_{32}$	$(D_{193}:C_{96})$ . $C_2$	all 32

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

Homology

Abelianization:	$C_{2} \times C_{192} \simeq C_{2} \times C_{64} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 6608 subgroups in 80 conjugacy classes, 46 normal (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $C_{193}:C_{32}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_2\times C_{192}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{1158}:C_{32}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{2316}$	$G/\operatorname{Fit} \simeq$ $C_{32}$
Radical:	$R \simeq$ $D_{193}:C_{192}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

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 74112: $D_{193}:C_{192}$

Order 37056: $C_{193}:C_{192}$ x 2, $D_{193}:C_{96}$

Order 24704: $D_{193}:C_{64}$

Order 18528: $C_{193}:C_{96}$ x 2, $D_{193}:C_{48}$

Order 12352: $C_{193}:C_{64}$ x 2, $D_{193}:C_{32}$

Order 9264: $C_{193}:C_{48}$ x 2, $D_{193}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 2, $D_{193}:C_{16}$

Order 4632: $C_{193}:C_{24}$ x 2, $C_{12}\times D_{193}$

Order 3088: $C_{193}:C_{16}$ x 2, $D_{193}:C_8$

Order 2316: $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 1544: $C_{193}:C_8$ x 2, $C_4\times D_{193}$

Order 1158: $C_3\times D_{193}$ x 2, $C_{1158}$

Order 772: $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 579: $C_{579}$

Order 386: $D_{193}$ x 2, $C_{386}$

Order 384: $C_2\times C_{192}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 2, $C_2\times C_{96}$

Order 128: $C_2\times C_{64}$

Order 96: $C_{96}$ x 2, $C_2\times C_{48}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 74112: $D_{193}:C_{192}$

Order 37056: $D_{193}:C_{96}$, $C_{193}:C_{192}$

Order 24704: $D_{193}:C_{64}$

Order 18528: $C_{193}:C_{96}$ x 2, $D_{193}:C_{48}$

Order 12352: $C_{193}:C_{64}$, $D_{193}:C_{32}$

Order 9264: $C_{193}:C_{48}$ x 2, $D_{193}:C_{24}$

Order 6176: $C_{193}:C_{32}$ x 2, $D_{193}:C_{16}$

Order 4632: $C_{193}:C_{24}$ x 2, $C_{12}\times D_{193}$

Order 3088: $C_{193}:C_{16}$ x 2, $D_{193}:C_8$

Order 2316: $C_3\times D_{386}$, $C_{193}:C_{12}$, $C_{2316}$

Order 1544: $C_{193}:C_8$ x 2, $C_4\times D_{193}$

Order 1158: $C_{1158}$, $C_3\times D_{193}$

Order 772: $D_{386}$, $C_{772}$, $C_{193}:C_4$

Order 579: $C_{579}$

Order 386: $C_{386}$, $D_{193}$

Order 384: $C_2\times C_{192}$

Order 193: $C_{193}$

Order 192: $C_{192}$, $C_2\times C_{96}$

Order 128: $C_2\times C_{64}$

Order 96: $C_{96}$ x 2, $C_2\times C_{48}$

Order 64: $C_2\times C_{32}$, $C_{64}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 74112: $D_{193}:C_{192}$ ($C_1$)

Order 37056: $C_{193}:C_{192}$ ($C_2$) x 2, $D_{193}:C_{96}$ ($C_2$)

Order 24704: $D_{193}:C_{64}$ ($C_3$)

Order 18528: $D_{193}:C_{48}$ ($C_4$), $C_{193}:C_{96}$ ($C_2^2$), $C_{193}:C_{96}$ ($C_4$)

Order 12352: $C_{193}:C_{64}$ ($C_6$) x 2, $D_{193}:C_{32}$ ($C_6$)

Order 9264: $C_{193}:C_{48}$ ($C_2\times C_4$), $C_{193}:C_{48}$ ($C_8$), $D_{193}:C_{24}$ ($C_8$)

Order 6176: $C_{193}:C_{32}$ ($C_2\times C_6$), $C_{193}:C_{32}$ ($C_{12}$), $D_{193}:C_{16}$ ($C_{12}$)

Order 4632: $C_{12}\times D_{193}$ ($C_{16}$), $C_{193}:C_{24}$ ($C_2\times C_8$), $C_{193}:C_{24}$ ($C_{16}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{12}$), $C_{193}:C_{16}$ ($C_{24}$), $D_{193}:C_8$ ($C_{24}$)

Order 2316: $C_3\times D_{386}$ ($C_{32}$), $C_{193}:C_{12}$ ($C_2\times C_{16}$), $C_{2316}$ ($C_{32}$)

Order 1544: $C_4\times D_{193}$ ($C_{48}$), $C_{193}:C_8$ ($C_2\times C_{24}$), $C_{193}:C_8$ ($C_{48}$)

Order 1158: $C_3\times D_{193}$ ($C_{64}$) x 2, $C_{1158}$ ($C_2\times C_{32}$)

Order 772: $D_{386}$ ($C_{96}$), $C_{772}$ ($C_{96}$), $C_{193}:C_4$ ($C_2\times C_{48}$)

Order 579: $C_{579}$ ($C_2\times C_{64}$)

Order 386: $D_{193}$ ($C_{192}$) x 2, $C_{386}$ ($C_2\times C_{96}$)

Order 193: $C_{193}$ ($C_2\times C_{192}$)

Order 12: $C_{12}$ ($C_{193}:C_{32}$)

Order 6: $C_6$ ($C_{386}:C_{32}$)

Order 4: $C_4$ ($C_{193}:C_{96}$)

Order 3: $C_3$ ($D_{193}:C_{64}$)

Order 2: $C_2$ ($C_{1158}:C_{32}$)

Order 1: $C_1$ ($D_{193}:C_{192}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 74112: $D_{193}:C_{192}$ ($C_1$)

Order 37056: $D_{193}:C_{96}$ ($C_2$), $C_{193}:C_{192}$ ($C_2$)

Order 24704: $D_{193}:C_{64}$ ($C_3$)

Order 18528: $D_{193}:C_{48}$ ($C_4$), $C_{193}:C_{96}$ ($C_2^2$), $C_{193}:C_{96}$ ($C_4$)

Order 12352: $C_{193}:C_{64}$ ($C_6$), $D_{193}:C_{32}$ ($C_6$)

Order 9264: $C_{193}:C_{48}$ ($C_2\times C_4$), $C_{193}:C_{48}$ ($C_8$), $D_{193}:C_{24}$ ($C_8$)

Order 6176: $C_{193}:C_{32}$ ($C_2\times C_6$), $C_{193}:C_{32}$ ($C_{12}$), $D_{193}:C_{16}$ ($C_{12}$)

Order 4632: $C_{12}\times D_{193}$ ($C_{16}$), $C_{193}:C_{24}$ ($C_2\times C_8$), $C_{193}:C_{24}$ ($C_{16}$)

Order 3088: $C_{193}:C_{16}$ ($C_2\times C_{12}$), $C_{193}:C_{16}$ ($C_{24}$), $D_{193}:C_8$ ($C_{24}$)

Order 2316: $C_3\times D_{386}$ ($C_{32}$), $C_{193}:C_{12}$ ($C_2\times C_{16}$), $C_{2316}$ ($C_{32}$)

Order 1544: $C_4\times D_{193}$ ($C_{48}$), $C_{193}:C_8$ ($C_2\times C_{24}$), $C_{193}:C_8$ ($C_{48}$)

Order 1158: $C_{1158}$ ($C_2\times C_{32}$), $C_3\times D_{193}$ ($C_{64}$)

Order 772: $D_{386}$ ($C_{96}$), $C_{772}$ ($C_{96}$), $C_{193}:C_4$ ($C_2\times C_{48}$)

Order 579: $C_{579}$ ($C_2\times C_{64}$)

Order 386: $C_{386}$ ($C_2\times C_{96}$), $D_{193}$ ($C_{192}$)

Order 193: $C_{193}$ ($C_2\times C_{192}$)

Order 12: $C_{12}$ ($C_{193}:C_{32}$)

Order 6: $C_6$ ($C_{386}:C_{32}$)

Order 4: $C_4$ ($C_{193}:C_{96}$)

Order 3: $C_3$ ($D_{193}:C_{64}$)

Order 2: $C_2$ ($C_{1158}:C_{32}$)

Order 1: $C_1$ ($D_{193}:C_{192}$)

Series

Derived series

$D_{193}:C_{192}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$D_{193}:C_{192}$

$\rhd$

$C_{193}:C_{192}$

$\rhd$

$C_{193}:C_{96}$

$\rhd$

$C_{193}:C_{48}$

$\rhd$

$C_{193}:C_{24}$

$\rhd$

$C_{193}:C_{12}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$D_{193}:C_{192}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $456 \times 456$ 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 $34 \times 34$ rational character table.