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G = C33⋊8D4⋊C2  order 432 = 24·33

8th semidirect product of C33⋊8D4 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C12.40(S32), (C3×Dic6)⋊8S3, C33⋊8D4⋊8C2, Dic6⋊5(C3⋊S3), (C3×C12).143D6, C33⋊15(C4○D4), C33⋊12D4⋊5C2, C3⋊Dic3.50D6, C3⋊3(D6.6D6), (C3×Dic3).15D6, C3⋊1(C12.26D6), C32⋊20(C4○D12), C32⋊7(Q8⋊3S3), (C32×Dic6)⋊12C2, (C32×C6).43C23, (C32×C12).45C22, (C32×Dic3).15C22, (C4×C3⋊S3)⋊7S3, C6.53(C2×S32), (C12×C3⋊S3)⋊6C2, C4.14(S3×C3⋊S3), C12.36(C2×C3⋊S3), (C2×C3⋊S3).43D6, C6.6(C22×C3⋊S3), C33⋊8(C2×C4)⋊5C2, Dic3.3(C2×C3⋊S3), (C6×C3⋊S3).52C22, (C3×C6).101(C22×S3), (C3×C3⋊Dic3).53C22, (C2×C33⋊C2).7C22, C2.10(C2×S3×C3⋊S3), SmallGroup(432,665)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊8D4⋊C2
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8(C2×C4) — C33⋊8D4⋊C2
Lower central
C33 — C32×C6 — C33⋊8D4⋊C2
Upper central
C1 — C2 — C4

Generators and relations for C33⋊8D4⋊C2
 G = < a,b,c,d,e,f | a3=b3=c3=d4=e2=f2=1, ab=ba, ac=ca, ad=da, eae=faf=a-1, bc=cb, bd=db, ebe=fbf=b-1, dcd-1=ece=fcf=c-1, ede=d-1, df=fd, fef=d2e >

Subgroups: 2104 in 304 conjugacy classes, 68 normal (22 characteristic)
C1, C2, C2 [×3], C3, C3 [×4], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×22], C6, C6 [×4], C6 [×5], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×4], C32 [×4], Dic3 [×2], Dic3 [×4], C12, C12 [×4], C12 [×13], D6 [×22], C2×C6, C4○D4, C3×S3 [×4], C3⋊S3 [×19], C3×C6, C3×C6 [×4], C3×C6 [×4], Dic6, C4×S3 [×14], D12 [×17], C3⋊D4 [×2], C2×C12, C3×Q8 [×4], C33, C3×Dic3 [×8], C3×Dic3 [×4], C3⋊Dic3, C3×C12, C3×C12 [×4], C3×C12 [×6], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×18], C4○D12, Q8⋊3S3 [×4], C3×C3⋊S3, C33⋊C2 [×2], C32×C6, C6.D6 [×8], C3⋊D12 [×8], C3×Dic6 [×4], S3×C12 [×4], C4×C3⋊S3, C4×C3⋊S3 [×2], C12⋊S3 [×11], Q8×C32, C32×Dic3 [×2], C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C2×C33⋊C2 [×2], D6.6D6 [×4], C12.26D6, C33⋊8(C2×C4) [×2], C33⋊8D4 [×2], C32×Dic6, C12×C3⋊S3, C33⋊12D4, C33⋊8D4⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], C23, D6 [×15], C4○D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], C4○D12, Q8⋊3S3 [×4], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, D6.6D6 [×4], C12.26D6, C2×S3×C3⋊S3, C33⋊8D4⋊C2

Smallest permutation representation of C33⋊8D4⋊C2
►On 72 points
Generators in S72
(1 37 11)(2 38 12)(3 39 9)(4 40 10)(5 62 28)(6 63 25)(7 64 26)(8 61 27)(13 31 60)(14 32 57)(15 29 58)(16 30 59)(17 42 50)(18 43 51)(19 44 52)(20 41 49)(21 66 45)(22 67 46)(23 68 47)(24 65 48)(33 70 55)(34 71 56)(35 72 53)(36 69 54)

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

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

(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)

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

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

54 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6E6F6G6H6I6J6K12A12B12C···12N12O···12V12W12X
order122223···33333444446···6666666121212···1212···121212
size111854542···24444266992···244441818224···412···121818

54 irreducible representations

dim111111222222224444
type++++++++++++++++
imageC1C2C2C2C2C2S3S3D6D6D6D6C4○D4C4○D12S32Q8⋊3S3C2×S32D6.6D6
kernelC33⋊8D4⋊C2C33⋊8(C2×C4)C33⋊8D4C32×Dic6C12×C3⋊S3C33⋊12D4C3×Dic6C4×C3⋊S3C3×Dic3C3⋊Dic3C3×C12C2×C3⋊S3C33C32C12C32C6C3
# reps122111418151244448

Matrix representation of C33⋊8D4⋊C2 ►in GL8(𝔽13)

110000000
111000000
00100000
00010000
00001000
00000100
000000012
000000112
,
110000000
111000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000001200
000011200
00000010
00000001
,
10000000
01000000
00730000
00560000
000001200
000012000
00000010
00000001
,
10000000
112000000
00100000
004120000
00000100
00001000
00000001
00000010
,
10000000
112000000
004110000
00190000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(13))| [1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C33⋊8D4⋊C2 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_8D_4\rtimes C_2
 
% in TeX
 
G:=Group("C3^3:8D4:C2");
 
// GroupNames label
 
G:=SmallGroup(432,665);
 
// by ID
 
G=gap.SmallGroup(432,665);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^4=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=f*a*f=a^-1,b*c=c*b,b*d=d*b,e*b*e=f*b*f=b^-1,d*c*d^-1=e*c*e=f*c*f=c^-1,e*d*e=d^-1,d*f=f*d,f*e*f=d^2*e>;
 
// generators/relations
 

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