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G = He33C8order 216 = 23·33

1st semidirect product of He3 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial

Aliases: He33C8, C322C24, (C3×C6).C12, C324C8⋊C3, C12.9(C3×S3), C322(C3⋊C8), (C3×C12).2C6, (C3×C12).5S3, C2.(C32⋊C12), (C2×He3).2C4, (C4×He3).3C2, (C3×C6).1Dic3, C6.2(C3×Dic3), C4.2(C32⋊C6), C3.2(C3×C3⋊C8), SmallGroup(216,14)

Series: Derived Chief Lower central Upper central

C1C32 — He33C8
C1C3C32C3×C6C3×C12C4×He3 — He33C8
C32 — He33C8
C1C4

Generators and relations for He33C8
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, dbd-1=b-1, cd=dc >

3C3
3C3
6C3
3C6
3C6
6C6
2C32
9C8
3C12
3C12
6C12
2C3×C6
3C3⋊C8
9C24
9C3⋊C8
2C3×C12
3C3×C3⋊C8

Smallest permutation representation of He33C8
On 72 points
Generators in S72
(1 68 11)(2 12 69)(3 70 13)(4 14 71)(5 72 15)(6 16 65)(7 66 9)(8 10 67)(17 30 43)(18 44 31)(19 32 45)(20 46 25)(21 26 47)(22 48 27)(23 28 41)(24 42 29)(33 56 62)(34 63 49)(35 50 64)(36 57 51)(37 52 58)(38 59 53)(39 54 60)(40 61 55)
(1 34 31)(2 32 35)(3 36 25)(4 26 37)(5 38 27)(6 28 39)(7 40 29)(8 30 33)(9 55 42)(10 43 56)(11 49 44)(12 45 50)(13 51 46)(14 47 52)(15 53 48)(16 41 54)(17 62 67)(18 68 63)(19 64 69)(20 70 57)(21 58 71)(22 72 59)(23 60 65)(24 66 61)
(1 11 18)(2 12 19)(3 13 20)(4 14 21)(5 15 22)(6 16 23)(7 9 24)(8 10 17)(25 46 57)(26 47 58)(27 48 59)(28 41 60)(29 42 61)(30 43 62)(31 44 63)(32 45 64)(33 56 67)(34 49 68)(35 50 69)(36 51 70)(37 52 71)(38 53 72)(39 54 65)(40 55 66)
(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)

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

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

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

He33C8 is a maximal subgroup of
C32⋊C6⋊C8  He3⋊M4(2)  C12.89S32  He33M4(2)  He33D8  He34SD16  He35SD16  He33Q16  C8×C32⋊C6  He35M4(2)  He37M4(2)  He38SD16  He36D8  He36Q16  He310SD16
He33C8 is a maximal quotient of
He33C16

40 conjugacy classes

class 1  2 3A3B3C3D3E3F4A4B6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G···12L24A···24H
order1233333344666666888812121212121212···1224···24
size112336661123366699992233336···69···9

40 irreducible representations

dim11111111222222666
type+++-+-
imageC1C2C3C4C6C8C12C24S3Dic3C3×S3C3⋊C8C3×Dic3C3×C3⋊C8C32⋊C6C32⋊C12He33C8
kernelHe33C8C4×He3C324C8C2×He3C3×C12He3C3×C6C32C3×C12C3×C6C12C32C6C3C4C2C1
# reps11222448112224112

Matrix representation of He33C8 in GL6(𝔽73)

001000
000100
000010
000001
100000
010000
,
010000
72720000
000100
00727200
000001
00007272
,
000001
00007272
100000
010000
00727200
001000
,
9201164920
116453621164
1164920920
536211641164
9209201164
116411645362

G:=sub<GL(6,GF(73))| [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],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,72,0,0,0,0,1,72,0,0,0,0],[9,11,11,53,9,11,20,64,64,62,20,64,11,53,9,11,9,11,64,62,20,64,20,64,9,11,9,11,11,53,20,64,20,64,64,62] >;

He33C8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3C_8
% in TeX

G:=Group("He3:3C8");
// GroupNames label

G:=SmallGroup(216,14);
// by ID

G=gap.SmallGroup(216,14);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,1450,5189]);
// Polycyclic

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

Export

Subgroup lattice of He33C8 in TeX

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