Copied to
clipboard

G = (C6×D4)⋊9C4order 192 = 26·3

5th semidirect product of C6×D4 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C6×D4)⋊9C4, (C6×Q8)⋊9C4, C4○D46Dic3, (C2×D4)⋊9Dic3, C4○D4.53D6, (C2×Q8)⋊9Dic3, C12.453(C2×D4), (C2×C12).198D4, D4.7(C2×Dic3), C12.86(C22×C4), Q8.12(C2×Dic3), (C4×Dic3)⋊7C22, (C22×C6).114D4, (C22×C4).178D6, Q83Dic310C2, C12.38(C22⋊C4), (C2×C12).482C23, C34(C42⋊C22), C23.38(C3⋊D4), C4.Dic324C22, C4.16(C22×Dic3), C4.23(C6.D4), C23.26D620C2, (C22×C12).208C22, C22.6(C6.D4), (C3×C4○D4)⋊4C4, (C6×C4○D4).5C2, (C2×C6).40(C2×D4), (C2×C4○D4).11S3, (C3×D4).24(C2×C4), C4.144(C2×C3⋊D4), C6.85(C2×C22⋊C4), (C3×Q8).25(C2×C4), (C2×C12).126(C2×C4), (C2×C4).90(C3⋊D4), (C2×C4.Dic3)⋊22C2, (C2×C4).29(C2×Dic3), C22.12(C2×C3⋊D4), (C2×C6).27(C22⋊C4), (C2×C4).567(C22×S3), C2.21(C2×C6.D4), (C3×C4○D4).42C22, SmallGroup(192,795)

Series: Derived Chief Lower central Upper central

C1C12 — (C6×D4)⋊9C4
C1C3C6C12C2×C12C4.Dic3C2×C4.Dic3 — (C6×D4)⋊9C4
C3C6C12 — (C6×D4)⋊9C4
C1C4C22×C4C2×C4○D4

Generators and relations for (C6×D4)⋊9C4
 G = < a,b,c,d | a6=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 328 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×5], C6, C6 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, Dic3 [×2], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×5], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×6], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×Dic3 [×2], C4⋊Dic3, C6.D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C42⋊C22, Q83Dic3 [×4], C2×C4.Dic3, C23.26D6, C6×C4○D4, (C6×D4)⋊9C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C42⋊C22, C2×C6.D4, (C6×D4)⋊9C4

Smallest permutation representation of (C6×D4)⋊9C4
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)
(1 19 41 9)(2 20 42 10)(3 21 37 11)(4 22 38 12)(5 23 39 7)(6 24 40 8)(13 44 29 35)(14 45 30 36)(15 46 25 31)(16 47 26 32)(17 48 27 33)(18 43 28 34)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 33)(8 34)(9 35)(10 36)(11 31)(12 32)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 44)(20 45)(21 46)(22 47)(23 48)(24 43)
(2 40)(3 5)(4 38)(6 42)(7 11)(8 20)(10 24)(12 22)(13 44 29 35)(14 34 30 43)(15 48 25 33)(16 32 26 47)(17 46 27 31)(18 36 28 45)(21 23)(37 39)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J4K6A6B6C6D···6I8A8B8C8D12A12B12C12D12E···12J
order12222223444444444446666···688881212121212···12
size112224421122244121212122224···41212121222224···4

42 irreducible representations

dim11111111222222222244
type+++++++++---+
imageC1C2C2C2C2C4C4C4S3D4D4D6Dic3Dic3Dic3D6C3⋊D4C3⋊D4C42⋊C22(C6×D4)⋊9C4
kernel(C6×D4)⋊9C4Q83Dic3C2×C4.Dic3C23.26D6C6×C4○D4C6×D4C6×Q8C3×C4○D4C2×C4○D4C2×C12C22×C6C22×C4C2×D4C2×Q8C4○D4C4○D4C2×C4C23C3C1
# reps14111224131111226224

Matrix representation of (C6×D4)⋊9C4 in GL4(𝔽73) generated by

603000
433000
006030
004330
,
270460
027046
00460
00046
,
7146659
5966147
14286659
4559147
,
072060
720600
00046
00460
G:=sub<GL(4,GF(73))| [60,43,0,0,30,30,0,0,0,0,60,43,0,0,30,30],[27,0,0,0,0,27,0,0,46,0,46,0,0,46,0,46],[7,59,14,45,14,66,28,59,66,14,66,14,59,7,59,7],[0,72,0,0,72,0,0,0,0,60,0,46,60,0,46,0] >;

(C6×D4)⋊9C4 in GAP, Magma, Sage, TeX

(C_6\times D_4)\rtimes_9C_4
% in TeX

G:=Group("(C6xD4):9C4");
// GroupNames label

G:=SmallGroup(192,795);
// by ID

G=gap.SmallGroup(192,795);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations

׿
×
𝔽