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## G = C24.3Dic3order 192 = 26·3

### 1st non-split extension by C24 of Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.3Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C12.55D4 — C24.3Dic3
 Lower central C3 — C6 — C2×C6 — C24.3Dic3
 Upper central C1 — C22 — C22×C4 — C2×C22⋊C4

Generators and relations for C24.3Dic3
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=ce3, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 248 in 98 conjugacy classes, 35 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×3], C22 [×3], C22 [×10], C6 [×3], C6 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×3], C23, C23 [×2], C23 [×4], C12 [×3], C2×C6 [×3], C2×C6 [×10], C22⋊C4 [×2], C2×C8 [×2], C22×C4 [×2], C24, C3⋊C8 [×2], C2×C12 [×2], C2×C12 [×3], C22×C6, C22×C6 [×2], C22×C6 [×4], C22⋊C8 [×2], C2×C22⋊C4, C2×C3⋊C8 [×2], C3×C22⋊C4 [×2], C22×C12 [×2], C23×C6, C23⋊C8, C12.55D4 [×2], C6×C22⋊C4, C24.3Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×C8, M4(2), C3⋊C8 [×2], C2×Dic3, C3⋊D4 [×2], C22⋊C8, C23⋊C4, C4.D4, C2×C3⋊C8, C4.Dic3, C6.D4, C23⋊C8, C12.55D4, C12.D4, C23.7D6, C24.3Dic3

Smallest permutation representation of C24.3Dic3
On 48 points
Generators in S48
(1 7)(2 37)(3 9)(4 39)(5 11)(6 41)(8 43)(10 45)(12 47)(13 19)(14 36)(15 21)(16 26)(17 23)(18 28)(20 30)(22 32)(24 34)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)
(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(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 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
(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 26 10 35 7 32 4 29)(2 31 11 28 8 25 5 34)(3 36 12 33 9 30 6 27)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A ··· 6G 6H 6I 6J 6K 8A ··· 8H 12A ··· 12H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 4 4 2 2 2 2 2 4 4 2 ··· 2 4 4 4 4 12 ··· 12 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - + + image C1 C2 C2 C4 C4 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 C23⋊C4 C4.D4 C12.D4 C23.7D6 kernel C24.3Dic3 C12.55D4 C6×C22⋊C4 C22×C12 C23×C6 C22×C6 C2×C22⋊C4 C2×C12 C22×C4 C22×C4 C24 C2×C6 C2×C4 C23 C22 C6 C6 C2 C2 # reps 1 2 1 2 2 8 1 2 1 1 1 2 4 4 4 1 1 2 2

Matrix representation of C24.3Dic3 in GL8(𝔽73)

 72 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 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 72 0 0 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 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 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 64 28 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0
,
 23 21 0 0 0 0 0 0 27 50 0 0 0 0 0 0 0 0 51 0 0 0 0 0 0 0 44 22 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[64,0,0,0,0,0,0,0,28,8,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[23,27,0,0,0,0,0,0,21,50,0,0,0,0,0,0,0,0,51,44,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C24.3Dic3 in GAP, Magma, Sage, TeX

C_2^4._3{\rm Dic}_3
% in TeX

G:=Group("C2^4.3Dic3");
// GroupNames label

G:=SmallGroup(192,84);
// by ID

G=gap.SmallGroup(192,84);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,100,1123,6278]);
// Polycyclic

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