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G = C6.C25order 192 = 26·3

14th non-split extension by C6 of C25 acting via C25/C24=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.14C25, D6.8C24, C12.49C24, D12.41C23, Dic3.9C24, Dic6.41C23, C4oD4:22D6, (C2xD4):47D6, (C2xQ8):39D6, D4oD12:13C2, Q8oD12:13C2, (C22xC4):37D6, (C2xC6).5C24, D4:6D6:11C2, (S3xD4):13C22, (C6xD4):54C22, C4.46(S3xC23), C2.15(S3xC24), (S3xQ8):15C22, (C6xQ8):47C22, C3:D4.2C23, C4oD12:27C22, (C2xD12):65C22, (C4xS3).38C23, Q8.15D6:9C2, C3:1(C2.C25), (C3xD4).30C23, D4.30(C22xS3), Q8.41(C22xS3), (C3xQ8).31C23, D4:2S3:14C22, (C2xC12).568C23, (C22xC12):29C22, Q8:3S3:14C22, (C2xDic6):76C22, C22.10(S3xC23), C23.152(C22xS3), (C22xC6).250C23, (C22xS3).142C23, (C2xDic3).168C23, (S3xC4oD4):6C2, (C2xC4oD4):19S3, (C6xC4oD4):16C2, (S3xC2xC4):35C22, (C2xC4oD12):39C2, (C3xC4oD4):22C22, (C2xC3:D4):55C22, (C2xC4).646(C22xS3), SmallGroup(192,1523)

Series: Derived Chief Lower central Upper central

C1C6 — C6.C25
C1C3C6D6C22xS3S3xC2xC4S3xC4oD4 — C6.C25
C3C6 — C6.C25
C1C4C2xC4oD4

Generators and relations for C6.C25
 G = < a,b,c,d,e,f | a6=b2=c2=e2=f2=1, d2=a3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, cd=dc, ece=a3c, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 1672 in 810 conjugacy classes, 443 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C2xC4oD4, C2xC4oD4, 2+ 1+4, 2- 1+4, C2xDic6, S3xC2xC4, C2xD12, C4oD12, S3xD4, D4:2S3, S3xQ8, Q8:3S3, C2xC3:D4, C22xC12, C6xD4, C6xQ8, C3xC4oD4, C2.C25, C2xC4oD12, D4:6D6, Q8.15D6, S3xC4oD4, D4oD12, Q8oD12, C6xC4oD4, C6.C25
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, C25, S3xC23, C2.C25, S3xC24, C6.C25

Smallest permutation representation of C6.C25
On 48 points
Generators in S48
(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)
(2 6)(3 5)(7 10)(8 9)(11 12)(14 18)(15 17)(19 22)(20 21)(23 24)(26 30)(27 29)(31 34)(32 33)(35 36)(38 42)(39 41)(43 46)(44 45)(47 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)
(1 28 4 25)(2 29 5 26)(3 30 6 27)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

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

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

54 conjugacy classes

class 1 2A2B···2H2I···2P 3 4A4B4C···4I4J···4Q6A6B6C6D···6I12A12B12C12D12E···12J
order122···22···23444···44···46666···61212121212···12
size112···26···62112···26···62224···422224···4

54 irreducible representations

dim111111112222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2S3D6D6D6D6C2.C25C6.C25
kernelC6.C25C2xC4oD12D4:6D6Q8.15D6S3xC4oD4D4oD12Q8oD12C6xC4oD4C2xC4oD4C22xC4C2xD4C2xQ8C4oD4C3C1
# reps166284411331824

Matrix representation of C6.C25 in GL6(F13)

0120000
1120000
0012000
0001200
0000120
0000012
,
0120000
1200000
001000
0001200
000010
0000012
,
100000
010000
000050
000005
008000
000800
,
1200000
0120000
005000
000500
000050
000005
,
100000
010000
000010
000001
001000
000100
,
100000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,5,0,0,0,0,0,0,5,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6.C25 in GAP, Magma, Sage, TeX

C_6.C_2^5
% in TeX

G:=Group("C6.C2^5");
// GroupNames label

G:=SmallGroup(192,1523);
// by ID

G=gap.SmallGroup(192,1523);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=e^2=f^2=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^3*b,c*d=d*c,e*c*e=a^3*c,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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