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## G = C62⋊5C4order 144 = 24·32

### 3rd semidirect product of C62 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62⋊5C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C62⋊5C4
 Lower central C32 — C3×C6 — C62⋊5C4
 Upper central C1 — C22 — C23

Generators and relations for C625C4
G = < a,b,c | a6=b6=c4=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

Subgroups: 250 in 102 conjugacy classes, 51 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C22, C22 [×2], C22 [×2], C6 [×12], C6 [×8], C2×C4 [×2], C23, C32, Dic3 [×8], C2×C6 [×12], C2×C6 [×8], C22⋊C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×8], C22×C6 [×4], C3⋊Dic3 [×2], C62, C62 [×2], C62 [×2], C6.D4 [×4], C2×C3⋊Dic3 [×2], C2×C62, C625C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], Dic3 [×8], D6 [×4], C22⋊C4, C3⋊S3, C2×Dic3 [×4], C3⋊D4 [×8], C3⋊Dic3 [×2], C2×C3⋊S3, C6.D4 [×4], C2×C3⋊Dic3, C327D4 [×2], C625C4

Smallest permutation representation of C625C4
On 72 points
Generators in S72
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)
(1 17 10 9 13 5)(2 18 11 7 14 6)(3 16 12 8 15 4)(19 33 36 29 23 25)(20 31 34 30 24 26)(21 32 35 28 22 27)(37 68 46 40 71 43)(38 69 47 41 72 44)(39 70 48 42 67 45)(49 61 56 52 64 59)(50 62 57 53 65 60)(51 63 58 54 66 55)
(1 44 32 60)(2 46 33 56)(3 48 31 58)(4 42 20 54)(5 38 21 50)(6 40 19 52)(7 43 23 59)(8 45 24 55)(9 47 22 57)(10 41 28 53)(11 37 29 49)(12 39 30 51)(13 69 27 62)(14 71 25 64)(15 67 26 66)(16 70 34 63)(17 72 35 65)(18 68 36 61)

G:=sub<Sym(72)| (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,17,10,9,13,5)(2,18,11,7,14,6)(3,16,12,8,15,4)(19,33,36,29,23,25)(20,31,34,30,24,26)(21,32,35,28,22,27)(37,68,46,40,71,43)(38,69,47,41,72,44)(39,70,48,42,67,45)(49,61,56,52,64,59)(50,62,57,53,65,60)(51,63,58,54,66,55), (1,44,32,60)(2,46,33,56)(3,48,31,58)(4,42,20,54)(5,38,21,50)(6,40,19,52)(7,43,23,59)(8,45,24,55)(9,47,22,57)(10,41,28,53)(11,37,29,49)(12,39,30,51)(13,69,27,62)(14,71,25,64)(15,67,26,66)(16,70,34,63)(17,72,35,65)(18,68,36,61)>;

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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,17,10,9,13,5)(2,18,11,7,14,6)(3,16,12,8,15,4)(19,33,36,29,23,25)(20,31,34,30,24,26)(21,32,35,28,22,27)(37,68,46,40,71,43)(38,69,47,41,72,44)(39,70,48,42,67,45)(49,61,56,52,64,59)(50,62,57,53,65,60)(51,63,58,54,66,55), (1,44,32,60)(2,46,33,56)(3,48,31,58)(4,42,20,54)(5,38,21,50)(6,40,19,52)(7,43,23,59)(8,45,24,55)(9,47,22,57)(10,41,28,53)(11,37,29,49)(12,39,30,51)(13,69,27,62)(14,71,25,64)(15,67,26,66)(16,70,34,63)(17,72,35,65)(18,68,36,61) );

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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,17,10,9,13,5),(2,18,11,7,14,6),(3,16,12,8,15,4),(19,33,36,29,23,25),(20,31,34,30,24,26),(21,32,35,28,22,27),(37,68,46,40,71,43),(38,69,47,41,72,44),(39,70,48,42,67,45),(49,61,56,52,64,59),(50,62,57,53,65,60),(51,63,58,54,66,55)], [(1,44,32,60),(2,46,33,56),(3,48,31,58),(4,42,20,54),(5,38,21,50),(6,40,19,52),(7,43,23,59),(8,45,24,55),(9,47,22,57),(10,41,28,53),(11,37,29,49),(12,39,30,51),(13,69,27,62),(14,71,25,64),(15,67,26,66),(16,70,34,63),(17,72,35,65),(18,68,36,61)])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6AB order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 size 1 1 1 1 2 2 2 2 2 2 18 18 18 18 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 kernel C62⋊5C4 C2×C3⋊Dic3 C2×C62 C62 C22×C6 C3×C6 C2×C6 C2×C6 C6 # reps 1 2 1 4 4 2 8 4 16

Matrix representation of C625C4 in GL4(𝔽13) generated by

 3 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 10 0 0 0 0 3 0 0 0 8 9
,
 0 1 0 0 12 0 0 0 0 0 2 8 0 0 1 11
G:=sub<GL(4,GF(13))| [3,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,3,8,0,0,0,9],[0,12,0,0,1,0,0,0,0,0,2,1,0,0,8,11] >;

C625C4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_5C_4
% in TeX

G:=Group("C6^2:5C4");
// GroupNames label

G:=SmallGroup(144,100);
// by ID

G=gap.SmallGroup(144,100);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=b^-1>;
// generators/relations

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