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G = C2×C23.7D6  order 192 = 26·3

Direct product of C2 and C23.7D6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.7D6, C24⋊6Dic3, C24.28D6, (C6×D4)⋊13C4, (C23×C6)⋊4C4, C6⋊2(C23⋊C4), (C22×C12)⋊6C4, (C2×D4)⋊5Dic3, (C2×D4).200D6, (C22×D4).6S3, C23⋊2(C2×Dic3), (C22×C4)⋊5Dic3, (C22×C6).108D4, (C6×D4).280C22, C23.71(C3⋊D4), C23.86(C22×S3), (C23×C6).45C22, C6.D4⋊43C22, (C22×C6).115C23, C22.6(C22×Dic3), C22.1(C6.D4), C3⋊3(C2×C23⋊C4), (C2×C12)⋊2(C2×C4), (D4×C2×C6).10C2, (C22×C6)⋊3(C2×C4), (C2×C4)⋊1(C2×Dic3), (C2×C6).38(C2×D4), C6.74(C2×C22⋊C4), (C2×C6.D4)⋊8C2, C22.10(C2×C3⋊D4), (C2×C6).195(C22×C4), C2.10(C2×C6.D4), (C2×C6).113(C22⋊C4), SmallGroup(192,778)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×C23.7D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C6.D4 — C2×C6.D4 — C2×C23.7D6
Lower central
C3 — C6 — C2×C6 — C2×C23.7D6
Upper central
C1 — C22 — C24 — C22×D4

Generators and relations for C2×C23.7D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 552 in 210 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C23⋊C4, C2×C22⋊C4, C22×D4, C6.D4, C6.D4, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C2×C23⋊C4, C23.7D6, C2×C6.D4, D4×C2×C6, C2×C23.7D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C23⋊C4, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C23⋊C4, C23.7D6, C2×C6.D4, C2×C23.7D6

Smallest permutation representation of C2×C23.7D6
►On 48 points
Generators in S48
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 26)(14 27)(15 28)(16 29)(17 30)(18 25)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)

(1 14)(2 39)(3 16)(4 41)(5 18)(6 37)(7 32)(8 24)(9 34)(10 20)(11 36)(12 22)(13 46)(15 48)(17 44)(19 29)(21 25)(23 27)(26 31)(28 33)(30 35)(38 47)(40 43)(42 45)

(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)

(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 19)(2 28 48 8)(3 23)(4 26 44 12)(5 21)(6 30 46 10)(7 16)(9 14)(11 18)(13 35 37 20)(15 33 39 24)(17 31 41 22)(25 42)(27 40)(29 38)(32 43)(34 47)(36 45)
 
G:=sub<Sym(48)| (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,14)(2,39)(3,16)(4,41)(5,18)(6,37)(7,32)(8,24)(9,34)(10,20)(11,36)(12,22)(13,46)(15,48)(17,44)(19,29)(21,25)(23,27)(26,31)(28,33)(30,35)(38,47)(40,43)(42,45), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,34)(20,35)(21,36)(22,31)(23,32)(24,33), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,19)(2,28,48,8)(3,23)(4,26,44,12)(5,21)(6,30,46,10)(7,16)(9,14)(11,18)(13,35,37,20)(15,33,39,24)(17,31,41,22)(25,42)(27,40)(29,38)(32,43)(34,47)(36,45)>;
 
G:=Group( (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,26)(14,27)(15,28)(16,29)(17,30)(18,25)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,14)(2,39)(3,16)(4,41)(5,18)(6,37)(7,32)(8,24)(9,34)(10,20)(11,36)(12,22)(13,46)(15,48)(17,44)(19,29)(21,25)(23,27)(26,31)(28,33)(30,35)(38,47)(40,43)(42,45), (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,34)(20,35)(21,36)(22,31)(23,32)(24,33), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,19)(2,28,48,8)(3,23)(4,26,44,12)(5,21)(6,30,46,10)(7,16)(9,14)(11,18)(13,35,37,20)(15,33,39,24)(17,31,41,22)(25,42)(27,40)(29,38)(32,43)(34,47)(36,45) );
 
G=PermutationGroup([[(1,23),(2,24),(3,19),(4,20),(5,21),(6,22),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,26),(14,27),(15,28),(16,29),(17,30),(18,25),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(1,14),(2,39),(3,16),(4,41),(5,18),(6,37),(7,32),(8,24),(9,34),(10,20),(11,36),(12,22),(13,46),(15,48),(17,44),(19,29),(21,25),(23,27),(26,31),(28,33),(30,35),(38,47),(40,43),(42,45)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,32),(8,33),(9,34),(10,35),(11,36),(12,31),(19,29),(20,30),(21,25),(22,26),(23,27),(24,28),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45)], [(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,34),(20,35),(21,36),(22,31),(23,32),(24,33)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,19),(2,28,48,8),(3,23),(4,26,44,12),(5,21),(6,30,46,10),(7,16),(9,14),(11,18),(13,35,37,20),(15,33,39,24),(17,31,41,22),(25,42),(27,40),(29,38),(32,43),(34,47),(36,45)]])
 

42 conjugacy classes

class 1 2A2B2C2D···2I2J2K 3 4A4B4C···4J6A···6G6H···6O12A12B12C12D
order12222···2223444···46···66···612121212
size11112···24424412···122···24···44444

42 irreducible representations

dim11111112222222244
type++++++--+-++
imageC1C2C2C2C4C4C4S3D4Dic3Dic3D6Dic3D6C3⋊D4C23⋊C4C23.7D6
kernelC2×C23.7D6C23.7D6C2×C6.D4D4×C2×C6C22×C12C6×D4C23×C6C22×D4C22×C6C22×C4C2×D4C2×D4C24C24C23C6C2
# reps14212421412211824

Matrix representation of C2×C23.7D6 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012000
000001200
000000120
000000012
,
10000000
01000000
00100000
00010000
00007300
000010600
000000610
00000037
,
120000000
012000000
001200000
000120000
00007300
000010600
00000073
000000106
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
01000000
10000000
0011110000
00290000
000000120
000000012
000012000
000001200
,
80000000
05000000
00800000
00850000
000012000
00009100
00000073
00000056

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,10,7],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,2,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,3,6] >;
 

C2×C23.7D6 in GAP, Magma, Sage, TeX 

C_2\times C_2^3._7D_6
 
% in TeX
 
G:=Group("C2xC2^3.7D6");
 
// GroupNames label
 
G:=SmallGroup(192,778);
 
// by ID
 
G=gap.SmallGroup(192,778);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,297,1684,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=c*b=b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^-1>;
 
// generators/relations
 

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