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G = C12.69S32order 432 = 24·33

26th non-split extension by C12 of S32 acting via S32/C3×S3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C12.69S32, C328(S3×C8), C3312(C2×C8), C33⋊C23C8, C324C817S3, (C3×C12).164D6, C335C4.3C4, C6.7(C6.D6), C31(C12.29D6), (C32×C12).66C22, C31(C8×C3⋊S3), (C3×C3⋊C8)⋊6S3, C3⋊C86(C3⋊S3), C6.1(C4×C3⋊S3), C4.24(S3×C3⋊S3), C12.39(C2×C3⋊S3), (C32×C3⋊C8)⋊11C2, (C3×C6).45(C4×S3), C2.1(C338(C2×C4)), (C3×C324C8)⋊10C2, (C32×C6).33(C2×C4), (C2×C33⋊C2).3C4, (C4×C33⋊C2).3C2, SmallGroup(432,432)

Series: Derived Chief Lower central Upper central

C1C33 — C12.69S32
C1C3C32C33C32×C6C32×C12C32×C3⋊C8 — C12.69S32
C33 — C12.69S32
C1C4

Generators and relations for C12.69S32
 G = < a,b,c,d,e | a3=b3=c8=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1160 in 196 conjugacy classes, 50 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C4, C22, S3 [×18], C6, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], Dic3 [×9], C12, C12 [×4], C12 [×4], D6 [×9], C2×C8, C3⋊S3 [×18], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C3⋊C8 [×4], C24 [×5], C4×S3 [×9], C33, C3⋊Dic3 [×9], C3×C12, C3×C12 [×4], C3×C12 [×4], C2×C3⋊S3 [×9], S3×C8 [×5], C33⋊C2 [×2], C32×C6, C3×C3⋊C8 [×4], C3×C3⋊C8 [×4], C324C8, C3×C24, C4×C3⋊S3 [×9], C335C4, C32×C12, C2×C33⋊C2, C12.29D6 [×4], C8×C3⋊S3, C32×C3⋊C8, C3×C324C8, C4×C33⋊C2, C12.69S32
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C8 [×2], C2×C4, D6 [×5], C2×C8, C3⋊S3, C4×S3 [×5], S32 [×4], C2×C3⋊S3, S3×C8 [×5], C6.D6 [×4], C4×C3⋊S3, S3×C3⋊S3, C12.29D6 [×4], C8×C3⋊S3, C338(C2×C4), C12.69S32

Smallest permutation representation of C12.69S32
On 72 points
Generators in S72
(1 31 68)(2 32 69)(3 25 70)(4 26 71)(5 27 72)(6 28 65)(7 29 66)(8 30 67)(9 22 56)(10 23 49)(11 24 50)(12 17 51)(13 18 52)(14 19 53)(15 20 54)(16 21 55)(33 62 45)(34 63 46)(35 64 47)(36 57 48)(37 58 41)(38 59 42)(39 60 43)(40 61 44)
(1 36 11)(2 12 37)(3 38 13)(4 14 39)(5 40 15)(6 16 33)(7 34 9)(8 10 35)(17 58 32)(18 25 59)(19 60 26)(20 27 61)(21 62 28)(22 29 63)(23 64 30)(24 31 57)(41 69 51)(42 52 70)(43 71 53)(44 54 72)(45 65 55)(46 56 66)(47 67 49)(48 50 68)
(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 24 48)(2 17 41)(3 18 42)(4 19 43)(5 20 44)(6 21 45)(7 22 46)(8 23 47)(9 63 66)(10 64 67)(11 57 68)(12 58 69)(13 59 70)(14 60 71)(15 61 72)(16 62 65)(25 52 38)(26 53 39)(27 54 40)(28 55 33)(29 56 34)(30 49 35)(31 50 36)(32 51 37)
(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I4A4B4C4D6A···6E6F6G6H6I8A8B8C8D8E8F8G8H12A···12J12K···12R24A···24P24Q24R24S24T
order12223···3333344446···666668888888812···1212···1224···2424242424
size1127272···244441127272···24444333399992···24···46···618181818

72 irreducible representations

dim111111122222444
type+++++++++
imageC1C2C2C2C4C4C8S3S3D6C4×S3S3×C8S32C6.D6C12.29D6
kernelC12.69S32C32×C3⋊C8C3×C324C8C4×C33⋊C2C335C4C2×C33⋊C2C33⋊C2C3×C3⋊C8C324C8C3×C12C3×C6C32C12C6C3
# reps11112284151020448

Matrix representation of C12.69S32 in GL6(𝔽73)

100000
010000
0007200
0017200
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
2200000
0220000
001000
000100
000001
000010
,
72720000
100000
0007200
0017200
000010
000001
,
100000
72720000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C12.69S32 in GAP, Magma, Sage, TeX

C_{12}._{69}S_3^2
% in TeX

G:=Group("C12.69S3^2");
// GroupNames label

G:=SmallGroup(432,432);
// by ID

G=gap.SmallGroup(432,432);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,36,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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