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## G = C62.111C23order 288 = 25·32

### 106th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.111C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.111C23
 Lower central C32 — C62 — C62.111C23
 Upper central C1 — C22 — C23

Generators and relations for C62.111C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=b3d >

Subgroups: 602 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×2], C6 [×6], C6 [×9], C2×C4 [×7], D4 [×2], C23, C23, C32, Dic3 [×9], C12 [×3], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×15], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6, C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×3], Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4, C6.D4, C6.D4 [×4], C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, S3×Dic3 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, C23.9D6, C23.23D6, D6⋊Dic3, Dic3⋊Dic3, C62.C22, C3×C6.D4, C625C4, C2×S3×Dic3, C6×C3⋊D4, C62.111C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3 [×2], C22.D4, S32, C4○D12, S3×D4, D42S3 [×3], C2×C3⋊D4, C2×S32, C23.9D6, C23.23D6, D6.3D6, D6.4D6, S3×C3⋊D4, C62.111C23

Smallest permutation representation of C62.111C23
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 16 3 18 5 14)(2 17 4 13 6 15)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 23 4 20)(2 22 5 19)(3 21 6 24)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 44 34 47)(32 43 35 46)(33 48 36 45)
(7 47)(8 48)(9 43)(10 44)(11 45)(12 46)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D6.3D6 D6.4D6 S3×C3⋊D4 kernel C62.111C23 D6⋊Dic3 Dic3⋊Dic3 C62.C22 C3×C6.D4 C62⋊5C4 C2×S3×Dic3 C6×C3⋊D4 C6.D4 C2×C3⋊D4 S3×C6 C2×Dic3 C22×S3 C22×C6 C3×C6 D6 C6 C23 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 1 2 4 4 4 1 1 3 1 2 2 2

Matrix representation of C62.111C23 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 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 12 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 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 12 12
,
 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

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

C62.111C23 in GAP, Magma, Sage, TeX

C_6^2._{111}C_2^3
% in TeX

G:=Group("C6^2.111C2^3");
// GroupNames label

G:=SmallGroup(288,617);
// by ID

G=gap.SmallGroup(288,617);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,1356,9414]);
// Polycyclic

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

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