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## G = C62.121C23order 288 = 25·32

### 116th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.121C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62.121C23
 Lower central C32 — C62 — C62.121C23
 Upper central C1 — C22 — C23

Generators and relations for C62.121C23
G = < a,b,c,d,e | a6=b6=c2=d2=e2=1, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ece=a3c, ede=a3b3d >

Subgroups: 1090 in 243 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×6], C22, C22 [×12], S3 [×6], C6 [×6], C6 [×9], C2×C4 [×3], D4 [×12], C23, C23 [×3], C32, Dic3 [×4], Dic3 [×6], C12 [×4], D6 [×16], C2×C6 [×2], C2×C6 [×17], C42, C2×D4 [×6], C3×S3 [×2], C3⋊S3, C3×C6, C3×C6 [×2], C3×C6, D12 [×4], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×20], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C22×C6 [×3], C41D4, C3×Dic3 [×4], C3⋊Dic3 [×2], S3×C6 [×6], C2×C3⋊S3 [×3], C62, C62 [×3], C4×Dic3 [×2], C2×D12 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4 [×7], C6×D4 [×2], D6⋊S3 [×2], C3⋊D12 [×4], C6×Dic3 [×2], C3×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×2], S3×C2×C6 [×2], C22×C3⋊S3, C2×C62, C123D4 [×2], Dic32, C2×D6⋊S3, C2×C3⋊D12 [×2], C6×C3⋊D4 [×2], C2×C327D4, C62.121C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], C3⋊D4 [×4], C22×S3 [×2], C41D4, S32, S3×D4 [×4], C2×C3⋊D4 [×2], C2×S32, C123D4 [×2], S3×C3⋊D4 [×2], Dic3⋊D6, C62.121C23

Smallest permutation representation of C62.121C23
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 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 36)(8 31)(9 32)(10 33)(11 34)(12 35)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6Q 6R 6S 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 4 12 12 36 2 2 4 6 6 6 6 18 18 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 S3×C3⋊D4 Dic3⋊D6 kernel C62.121C23 Dic32 C2×D6⋊S3 C2×C3⋊D12 C6×C3⋊D4 C2×C32⋊7D4 C2×C3⋊D4 C3×Dic3 C3⋊Dic3 C2×Dic3 C22×S3 C22×C6 Dic3 C23 C6 C22 C2 C2 # reps 1 1 1 2 2 1 2 4 2 2 2 2 8 1 4 1 4 2

Matrix representation of C62.121C23 in GL8(ℤ)

 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 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 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 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 1 1 0 0 0 0 0 0 0 0 1 2 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 0 1 0 0 0 0 0 0 1 0
,
 -1 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -2 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 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,Integers())| [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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-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,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C62.121C23 in GAP, Magma, Sage, TeX

C_6^2._{121}C_2^3
% in TeX

G:=Group("C6^2.121C2^3");
// GroupNames label

G:=SmallGroup(288,627);
// by ID

G=gap.SmallGroup(288,627);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,303,1356,9414]);
// Polycyclic

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

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