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G = C42.144D6  order 192 = 26·3

144th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.144D6, C6.922- (1+4), C6.1282+ (1+4), D6⋊3Q8⋊33C2, C12⋊2Q8⋊31C2, (Q8×Dic3)⋊21C2, (D4×Dic3)⋊32C2, (C2×D4).177D6, C4.4D4⋊15S3, (C2×Q8).164D6, C22⋊C4.37D6, C42⋊2S3⋊21C2, D6⋊3D4.13C2, C2.52(D4○D12), (C2×C6).226C24, (C2×C12).82C23, C2.53(Q8○D12), C12.127(C4○D4), C4.39(D4⋊2S3), (C4×C12).189C22, D6⋊C4.111C22, (C6×D4).159C22, C23.8D6⋊42C2, C23.58(C22×S3), (C22×C6).56C23, (C6×Q8).130C22, C23.11D6⋊42C2, Dic3.D4⋊42C2, C22.D12⋊27C2, Dic3⋊C4.49C22, (C22×S3).98C23, C4⋊Dic3.236C22, C22.247(S3×C23), (C2×Dic6).38C22, C3⋊9(C22.36C24), (C2×Dic3).116C23, (C4×Dic3).136C22, C6.D4.59C22, (C22×Dic3).146C22, C6.94(C2×C4○D4), (C3×C4.4D4)⋊18C2, C2.58(C2×D4⋊2S3), (S3×C2×C4).122C22, (C2×C4).199(C22×S3), (C2×C3⋊D4).64C22, (C3×C22⋊C4).68C22, SmallGroup(192,1241)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42.144D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C42⋊2S3 — C42.144D6
Lower central
C3 — C2×C6 — C42.144D6

Subgroups: 528 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×11], C22, C22 [×9], S3, C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, Dic3 [×7], C12 [×2], C12 [×4], D6 [×3], C2×C6, C2×C6 [×6], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4⋊Dic3 [×4], D6⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], S3×C2×C4, C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, C22.36C24, C12⋊2Q8, C42⋊2S3, Dic3.D4 [×2], C23.8D6 [×2], C23.11D6 [×2], C22.D12 [×2], D4×Dic3, D6⋊3D4, Q8×Dic3, D6⋊3Q8, C3×C4.4D4, C42.144D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), D4⋊2S3 [×2], S3×C23, C22.36C24, C2×D4⋊2S3, D4○D12, Q8○D12, C42.144D6

Generators and relations
 G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=b2c-1 >

Smallest permutation representation
►On 96 points
Generators in S96
(1 64 19 67)(2 68 20 65)(3 66 21 69)(4 70 22 61)(5 62 23 71)(6 72 24 63)(7 57 36 81)(8 82 31 58)(9 59 32 83)(10 84 33 60)(11 55 34 79)(12 80 35 56)(13 38 86 54)(14 49 87 39)(15 40 88 50)(16 51 89 41)(17 42 90 52)(18 53 85 37)(25 76 45 93)(26 94 46 77)(27 78 47 95)(28 96 48 73)(29 74 43 91)(30 92 44 75)

(1 43 9 50)(2 41 10 30)(3 45 11 52)(4 37 12 26)(5 47 7 54)(6 39 8 28)(13 62 95 57)(14 82 96 72)(15 64 91 59)(16 84 92 68)(17 66 93 55)(18 80 94 70)(19 29 32 40)(20 51 33 44)(21 25 34 42)(22 53 35 46)(23 27 36 38)(24 49 31 48)(56 77 61 85)(58 73 63 87)(60 75 65 89)(67 74 83 88)(69 76 79 90)(71 78 81 86)

(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)

(1 6 9 8)(2 7 10 5)(3 4 11 12)(13 89 95 75)(14 74 96 88)(15 87 91 73)(16 78 92 86)(17 85 93 77)(18 76 94 90)(19 24 32 31)(20 36 33 23)(21 22 34 35)(25 46 42 53)(26 52 37 45)(27 44 38 51)(28 50 39 43)(29 48 40 49)(30 54 41 47)(55 56 66 61)(57 60 62 65)(58 64 63 59)(67 72 83 82)(68 81 84 71)(69 70 79 80)
 
G:=sub<Sym(96)| (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,57,36,81)(8,82,31,58)(9,59,32,83)(10,84,33,60)(11,55,34,79)(12,80,35,56)(13,38,86,54)(14,49,87,39)(15,40,88,50)(16,51,89,41)(17,42,90,52)(18,53,85,37)(25,76,45,93)(26,94,46,77)(27,78,47,95)(28,96,48,73)(29,74,43,91)(30,92,44,75), (1,43,9,50)(2,41,10,30)(3,45,11,52)(4,37,12,26)(5,47,7,54)(6,39,8,28)(13,62,95,57)(14,82,96,72)(15,64,91,59)(16,84,92,68)(17,66,93,55)(18,80,94,70)(19,29,32,40)(20,51,33,44)(21,25,34,42)(22,53,35,46)(23,27,36,38)(24,49,31,48)(56,77,61,85)(58,73,63,87)(60,75,65,89)(67,74,83,88)(69,76,79,90)(71,78,81,86), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,6,9,8)(2,7,10,5)(3,4,11,12)(13,89,95,75)(14,74,96,88)(15,87,91,73)(16,78,92,86)(17,85,93,77)(18,76,94,90)(19,24,32,31)(20,36,33,23)(21,22,34,35)(25,46,42,53)(26,52,37,45)(27,44,38,51)(28,50,39,43)(29,48,40,49)(30,54,41,47)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,83,82)(68,81,84,71)(69,70,79,80)>;
 
G:=Group( (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,57,36,81)(8,82,31,58)(9,59,32,83)(10,84,33,60)(11,55,34,79)(12,80,35,56)(13,38,86,54)(14,49,87,39)(15,40,88,50)(16,51,89,41)(17,42,90,52)(18,53,85,37)(25,76,45,93)(26,94,46,77)(27,78,47,95)(28,96,48,73)(29,74,43,91)(30,92,44,75), (1,43,9,50)(2,41,10,30)(3,45,11,52)(4,37,12,26)(5,47,7,54)(6,39,8,28)(13,62,95,57)(14,82,96,72)(15,64,91,59)(16,84,92,68)(17,66,93,55)(18,80,94,70)(19,29,32,40)(20,51,33,44)(21,25,34,42)(22,53,35,46)(23,27,36,38)(24,49,31,48)(56,77,61,85)(58,73,63,87)(60,75,65,89)(67,74,83,88)(69,76,79,90)(71,78,81,86), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,6,9,8)(2,7,10,5)(3,4,11,12)(13,89,95,75)(14,74,96,88)(15,87,91,73)(16,78,92,86)(17,85,93,77)(18,76,94,90)(19,24,32,31)(20,36,33,23)(21,22,34,35)(25,46,42,53)(26,52,37,45)(27,44,38,51)(28,50,39,43)(29,48,40,49)(30,54,41,47)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,83,82)(68,81,84,71)(69,70,79,80) );
 
G=PermutationGroup([(1,64,19,67),(2,68,20,65),(3,66,21,69),(4,70,22,61),(5,62,23,71),(6,72,24,63),(7,57,36,81),(8,82,31,58),(9,59,32,83),(10,84,33,60),(11,55,34,79),(12,80,35,56),(13,38,86,54),(14,49,87,39),(15,40,88,50),(16,51,89,41),(17,42,90,52),(18,53,85,37),(25,76,45,93),(26,94,46,77),(27,78,47,95),(28,96,48,73),(29,74,43,91),(30,92,44,75)], [(1,43,9,50),(2,41,10,30),(3,45,11,52),(4,37,12,26),(5,47,7,54),(6,39,8,28),(13,62,95,57),(14,82,96,72),(15,64,91,59),(16,84,92,68),(17,66,93,55),(18,80,94,70),(19,29,32,40),(20,51,33,44),(21,25,34,42),(22,53,35,46),(23,27,36,38),(24,49,31,48),(56,77,61,85),(58,73,63,87),(60,75,65,89),(67,74,83,88),(69,76,79,90),(71,78,81,86)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,6,9,8),(2,7,10,5),(3,4,11,12),(13,89,95,75),(14,74,96,88),(15,87,91,73),(16,78,92,86),(17,85,93,77),(18,76,94,90),(19,24,32,31),(20,36,33,23),(21,22,34,35),(25,46,42,53),(26,52,37,45),(27,44,38,51),(28,50,39,43),(29,48,40,49),(30,54,41,47),(55,56,66,61),(57,60,62,65),(58,64,63,59),(67,72,83,82),(68,81,84,71),(69,70,79,80)])
 

Matrix representation ►G ⊆ GL8(𝔽13)

120000000
012000000
00100000
00010000
00003700
000061000
00000037
000000610
,
80000000
05000000
00100000
00010000
00000010
00000001
00001000
00000100
,
01000000
10000000
000120000
001120000
00000100
00001000
000000012
000000120
,
012000000
10000000
001120000
000120000
00000100
00001000
00000001
00000010

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

36 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J4K···4O6A6B6C6D6E12A···12F12G12H
order1222222344444444444···46666612···121212
size111144122224444666612···12222884···488

36 irreducible representations

dim11111111111122222244444
type++++++++++++++++++--+-
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ (1+4)2- (1+4)D4⋊2S3D4○D12Q8○D12
kernelC42.144D6C12⋊2Q8C42⋊2S3Dic3.D4C23.8D6C23.11D6C22.D12D4×Dic3D6⋊3D4Q8×Dic3D6⋊3Q8C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8C12C6C6C4C2C2
# reps11122221111111411411222

In GAP, Magma, Sage, TeX 

C_4^2._{144}D_6
 
% in TeX
 
G:=Group("C4^2.144D6");
 
// GroupNames label
 
G:=SmallGroup(192,1241);
 
// by ID
 
G=gap.SmallGroup(192,1241);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,675,570,409,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

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