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G = C23⋊C45S3order 192 = 26·3

The semidirect product of C23⋊C4 and S3 acting through Inn(C23⋊C4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊C45S3, (C2×Dic6)⋊4C4, (C2×D4).119D6, C23.10(C4×S3), C22⋊C4.43D6, (C6×D4).7C22, C22.23(S3×D4), (C22×Dic3)⋊4C4, C23.6D63C2, C23.7D63C2, (C22×C6).1C23, (C2×Dic3).132D4, C23.11(C22×S3), C23.16D623C2, Dic3.3(C22⋊C4), C31(C23.C23), C6.D4.1C22, (C22×Dic3).25C22, (S3×C2×C4)⋊1C4, (C2×C3⋊D4)⋊1C4, (C2×C4).5(C4×S3), (C3×C23⋊C4)⋊3C2, (C2×C12).5(C2×C4), (C2×C6).16(C2×D4), C22.12(S3×C2×C4), C2.11(S3×C22⋊C4), C6.10(C2×C22⋊C4), (C22×C6).5(C2×C4), (C2×C6).6(C22×C4), (C2×D42S3).1C2, (C22×S3).1(C2×C4), (C2×Dic3).1(C2×C4), (C2×C3⋊D4).1C22, (C3×C22⋊C4).82C22, SmallGroup(192,299)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23⋊C45S3
C1C3C6C2×C6C22×C6C22×Dic3C2×D42S3 — C23⋊C45S3
C3C6C2×C6 — C23⋊C45S3
C1C2C23C23⋊C4

Generators and relations for C23⋊C45S3
 G = < a,b,c,d,e,f | a2=b2=c2=d4=e3=f2=1, ab=ba, faf=ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=bcd, fef=e-1 >

Subgroups: 448 in 158 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2 [×5], C3, C4 [×10], C22, C22 [×2], C22 [×5], S3, C6, C6 [×4], C2×C4, C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×2], C23, Dic3 [×2], Dic3 [×2], Dic3 [×3], C12 [×3], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×2], C22×S3, C22×C6 [×2], C23⋊C4, C23⋊C4 [×3], C42⋊C2 [×2], C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], C2×Dic6, S3×C2×C4, D42S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C23.C23, C23.6D6 [×2], C23.7D6, C3×C23⋊C4, C23.16D6 [×2], C2×D42S3, C23⋊C45S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C23.C23, S3×C22⋊C4, C23⋊C45S3

Smallest permutation representation of C23⋊C45S3
On 48 points
Generators in S48
(1 6)(2 5)(3 35)(4 34)(7 26)(8 25)(9 46)(10 45)(11 18)(12 17)(13 44)(14 43)(15 22)(16 21)(19 39)(20 38)(23 32)(24 31)(27 33)(28 36)(29 42)(30 41)(37 47)(40 48)
(1 3)(2 26)(4 28)(5 7)(6 35)(8 33)(9 11)(10 37)(12 39)(13 15)(14 30)(16 32)(17 19)(18 46)(20 48)(21 23)(22 44)(24 42)(25 27)(29 31)(34 36)(38 40)(41 43)(45 47)
(1 27)(2 28)(3 25)(4 26)(5 36)(6 33)(7 34)(8 35)(9 38)(10 39)(11 40)(12 37)(13 31)(14 32)(15 29)(16 30)(17 47)(18 48)(19 45)(20 46)(21 41)(22 42)(23 43)(24 44)
(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 43 9)(2 44 10)(3 41 11)(4 42 12)(5 13 45)(6 14 46)(7 15 47)(8 16 48)(17 34 29)(18 35 30)(19 36 31)(20 33 32)(21 40 25)(22 37 26)(23 38 27)(24 39 28)
(1 2)(3 26)(4 25)(5 33)(6 36)(7 8)(9 44)(10 43)(11 22)(12 21)(13 20)(14 19)(15 48)(16 47)(17 30)(18 29)(23 39)(24 38)(27 28)(31 46)(32 45)(34 35)(37 41)(40 42)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C···4G4H4I4J4K···4O6A6B6C6D6E12A···12E
order12222223444···44444···46666612···12
size112224122334···466612···12244488···8

33 irreducible representations

dim1111111111222222448
type+++++++++++-
imageC1C2C2C2C2C2C4C4C4C4S3D4D6D6C4×S3C4×S3S3×D4C23.C23C23⋊C45S3
kernelC23⋊C45S3C23.6D6C23.7D6C3×C23⋊C4C23.16D6C2×D42S3C2×Dic6S3×C2×C4C22×Dic3C2×C3⋊D4C23⋊C4C2×Dic3C22⋊C4C2×D4C2×C4C23C22C3C1
# reps1211212222142122221

Matrix representation of C23⋊C45S3 in GL6(𝔽13)

1200000
0120000
000850
000805
008805
000305
,
100000
010000
0001112
0010112
000010
0000212
,
100000
010000
0012000
0001200
0000120
0000012
,
500000
050000
0012010
0012000
0012100
0011001
,
0120000
1120000
001000
000100
000010
000001
,
0120000
1200000
0012000
00120121
00121201
0011001

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

C23⋊C45S3 in GAP, Magma, Sage, TeX

C_2^3\rtimes C_4\rtimes_5S_3
% in TeX

G:=Group("C2^3:C4:5S3");
// GroupNames label

G:=SmallGroup(192,299);
// by ID

G=gap.SmallGroup(192,299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,58,570,438,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^3=f^2=1,a*b=b*a,f*a*f=a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b*c*d,f*e*f=e^-1>;
// generators/relations

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