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G = C334Q8order 216 = 23·33

2nd semidirect product of C33 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial

Aliases: C334Q8, C325Dic6, C6.15S32, (C3×C6).34D6, Dic3.(C3⋊S3), C3⋊Dic3.3S3, C31(C322Q8), C335C4.1C2, (C3×Dic3).3S3, C31(C324Q8), (C32×C6).12C22, (C32×Dic3).1C2, C6.7(C2×C3⋊S3), C2.7(S3×C3⋊S3), (C3×C3⋊Dic3).2C2, SmallGroup(216,130)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C334Q8
C1C3C32C33C32×C6C32×Dic3 — C334Q8
C33C32×C6 — C334Q8
C1C2

Generators and relations for C334Q8
 G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 396 in 96 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C3, C3 [×4], C3 [×4], C4 [×3], C6, C6 [×4], C6 [×4], Q8, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×13], C12 [×5], C3×C6, C3×C6 [×4], C3×C6 [×4], Dic6 [×5], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3⋊Dic3 [×9], C3×C12, C32×C6, C322Q8 [×4], C324Q8, C32×Dic3, C3×C3⋊Dic3, C335C4, C334Q8
Quotients: C1, C2 [×3], C22, S3 [×5], Q8, D6 [×5], C3⋊S3, Dic6 [×5], S32 [×4], C2×C3⋊S3, C322Q8 [×4], C324Q8, S3×C3⋊S3, C334Q8

Smallest permutation representation of C334Q8
On 72 points
Generators in S72
(1 15 33)(2 34 16)(3 13 35)(4 36 14)(5 31 60)(6 57 32)(7 29 58)(8 59 30)(9 54 63)(10 64 55)(11 56 61)(12 62 53)(17 72 65)(18 66 69)(19 70 67)(20 68 71)(21 40 49)(22 50 37)(23 38 51)(24 52 39)(25 41 48)(26 45 42)(27 43 46)(28 47 44)
(1 40 7)(2 8 37)(3 38 5)(4 6 39)(9 46 66)(10 67 47)(11 48 68)(12 65 45)(13 51 31)(14 32 52)(15 49 29)(16 30 50)(17 42 62)(18 63 43)(19 44 64)(20 61 41)(21 58 33)(22 34 59)(23 60 35)(24 36 57)(25 71 56)(26 53 72)(27 69 54)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)

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

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

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

C334Q8 is a maximal subgroup of
S3×C322Q8  C335(C2×Q8)  D6.S32  D6.4S32  S3×C324Q8  C3⋊S3×Dic6  C329(S3×Q8)  C12.73S32  C62.90D6  C62.91D6  C62.93D6
C334Q8 is a maximal quotient of
C62.80D6  C62.81D6  C62.82D6

33 conjugacy classes

class 1  2 3A···3E3F3G3H3I4A4B4C6A···6E6F6G6H6I12A···12H12I12J
order123···333334446···6666612···121212
size112···24444618542···244446···61818

33 irreducible representations

dim11112222244
type++++++-+-+-
imageC1C2C2C2S3S3Q8D6Dic6S32C322Q8
kernelC334Q8C32×Dic3C3×C3⋊Dic3C335C4C3×Dic3C3⋊Dic3C33C3×C6C32C6C3
# reps111141151044

Matrix representation of C334Q8 in GL6(𝔽13)

300000
090000
001000
000100
0000012
0000112
,
300000
090000
001000
000100
000010
000001
,
100000
010000
00121200
001000
000010
000001
,
010000
1200000
0012000
0001200
000001
000010
,
500000
080000
001000
00121200
000010
000001

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

C334Q8 in GAP, Magma, Sage, TeX

C_3^3\rtimes_4Q_8
% in TeX

G:=Group("C3^3:4Q8");
// GroupNames label

G:=SmallGroup(216,130);
// by ID

G=gap.SmallGroup(216,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,201,730,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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