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G = C24.35D6order 192 = 26·3

24th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.35D6, C6.22+ 1+4, C234(C4×S3), (C22×C4)⋊6D6, C22⋊C451D6, C6.9(C23×C4), D6⋊C457C22, (C2×C6).30C24, D6.2(C22×C4), C2.1(D46D6), Dic34D439C2, (C2×C12).571C23, Dic3⋊C458C22, (C22×C12)⋊34C22, C31(C22.11C24), (C4×Dic3)⋊46C22, (C23×C6).56C22, C22.19(S3×C23), Dic3.3(C22×C4), C6.D467C22, C23.16D624C2, (S3×C23).30C22, C23.230(C22×S3), (C22×C6).122C23, (C22×Dic3)⋊5C22, (C22×S3).150C23, (C2×Dic3).177C23, C3⋊D49(C2×C4), (C2×C3⋊D4)⋊9C4, (C4×C3⋊D4)⋊33C2, (C2×C22⋊C4)⋊7S3, (S3×C2×C4)⋊39C22, C2.11(S3×C22×C4), C22.24(S3×C2×C4), (C6×C22⋊C4)⋊26C2, (S3×C22⋊C4)⋊23C2, (C22×S3)⋊6(C2×C4), (C22×C6)⋊10(C2×C4), (C2×Dic3)⋊10(C2×C4), (C2×C6).18(C22×C4), (C22×C3⋊D4).9C2, (C2×C6.D4)⋊15C2, (C3×C22⋊C4)⋊61C22, (C2×C4).257(C22×S3), (C2×C3⋊D4).88C22, SmallGroup(192,1045)

Series: Derived Chief Lower central Upper central

C1C6 — C24.35D6
C1C3C6C2×C6C22×S3S3×C23C22×C3⋊D4 — C24.35D6
C3C6 — C24.35D6
C1C22C2×C22⋊C4

Generators and relations for C24.35D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 840 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×12], C22, C22 [×6], C22 [×22], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×16], C23 [×3], C23 [×4], C23 [×10], Dic3 [×4], Dic3 [×4], C12 [×4], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C2×C6 [×10], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×12], C24, C24, C4×S3 [×4], C2×Dic3 [×10], C2×Dic3 [×2], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×2], C22×S3 [×6], C22×S3 [×2], C22×C6 [×3], C22×C6 [×4], C22×C6 [×2], C2×C22⋊C4, C2×C22⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×12], C22×C12 [×2], S3×C23, C23×C6, C22.11C24, C23.16D6 [×2], S3×C22⋊C4 [×2], Dic34D4 [×4], C4×C3⋊D4 [×4], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.35D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4 [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D46D6 [×2], C24.35D6

Smallest permutation representation of C24.35D6
On 48 points
Generators in S48
(2 46)(4 48)(6 38)(8 40)(10 42)(12 44)(14 31)(16 33)(18 35)(20 25)(22 27)(24 29)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(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)
(1 45)(2 46)(3 47)(4 48)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 35 19 29)(14 28 20 34)(15 33 21 27)(16 26 22 32)(17 31 23 25)(18 36 24 30)(37 46 43 40)(38 39 44 45)(41 42 47 48)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M 3 4A···4H4I···4T6A···6G6H6I6J6K12A···12H
order12222···2222234···44···46···6666612···12
size11112···2666622···26···62···244444···4

54 irreducible representations

dim1111111112222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4S3D6D6D6C4×S32+ 1+4D46D6
kernelC24.35D6C23.16D6S3×C22⋊C4Dic34D4C4×C3⋊D4C2×C6.D4C6×C22⋊C4C22×C3⋊D4C2×C3⋊D4C2×C22⋊C4C22⋊C4C22×C4C24C23C6C2
# reps12244111161421824

Matrix representation of C24.35D6 in GL6(𝔽13)

1200000
0120000
001000
000100
0080120
0008012
,
100000
010000
0011900
004200
0000119
000042
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
880000
500000
005522
0080110
000088
000050
,
550000
080000
005522
0008011
000088
000005

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,8,0,0,0,0,1,0,8,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,5,0,0,0,0,8,0,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,2,11,8,5,0,0,2,0,8,0],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,2,0,8,0,0,0,2,11,8,5] >;

C24.35D6 in GAP, Magma, Sage, TeX

C_2^4._{35}D_6
% in TeX

G:=Group("C2^4.35D6");
// GroupNames label

G:=SmallGroup(192,1045);
// by ID

G=gap.SmallGroup(192,1045);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations

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