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## G = C4⋊C4.150D6order 192 = 26·3

### 23rd non-split extension by C4⋊C4 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4⋊C4.150D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×Q8⋊3S3 — C4⋊C4.150D6
 Lower central C3 — C6 — C12 — C4⋊C4.150D6
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for C4⋊C4.150D6
G = < a,b,c,d | a4=b4=d2=1, c6=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab-1, dcd=a2c5 >

Subgroups: 456 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, D4⋊C4, Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, S3×C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q83S3, Q83S3, C6×Q8, C23.24D4, C6.D8, C2.D24, Q82Dic3, C3×Q8⋊C4, C4⋊C47S3, S3×C2×C8, C2×Q83S3, C4⋊C4.150D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C4○D8, S3×C2×C4, S3×D4, C23.24D4, S3×C22⋊C4, Q8.7D6, D24⋊C2, C4⋊C4.150D6

Smallest permutation representation of C4⋊C4.150D6
On 96 points
Generators in S96
(1 82 7 76)(2 77 8 83)(3 84 9 78)(4 79 10 73)(5 74 11 80)(6 81 12 75)(13 29 19 35)(14 36 20 30)(15 31 21 25)(16 26 22 32)(17 33 23 27)(18 28 24 34)(37 58 43 52)(38 53 44 59)(39 60 45 54)(40 55 46 49)(41 50 47 56)(42 57 48 51)(61 88 67 94)(62 95 68 89)(63 90 69 96)(64 85 70 91)(65 92 71 86)(66 87 72 93)
(1 39 22 86)(2 66 23 49)(3 41 24 88)(4 68 13 51)(5 43 14 90)(6 70 15 53)(7 45 16 92)(8 72 17 55)(9 47 18 94)(10 62 19 57)(11 37 20 96)(12 64 21 59)(25 44 75 91)(26 65 76 60)(27 46 77 93)(28 67 78 50)(29 48 79 95)(30 69 80 52)(31 38 81 85)(32 71 82 54)(33 40 83 87)(34 61 84 56)(35 42 73 89)(36 63 74 58)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)(25 29)(26 28)(30 36)(31 35)(32 34)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 72)(47 71)(48 70)(49 87)(50 86)(51 85)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(73 81)(74 80)(75 79)(76 78)(82 84)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 6 6 12 12 2 2 2 3 3 3 3 4 4 4 4 12 12 2 2 2 2 2 2 2 6 6 6 6 4 4 8 8 8 8 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C4○D8 S3×D4 S3×D4 Q8.7D6 D24⋊C2 kernel C4⋊C4.150D6 C6.D8 C2.D24 Q8⋊2Dic3 C3×Q8⋊C4 C4⋊C4⋊7S3 S3×C2×C8 C2×Q8⋊3S3 Q8⋊3S3 Q8⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×Q8 Q8 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 8 1 1 2 2

Matrix representation of C4⋊C4.150D6 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 0 72 0 0 1 0
,
 46 0 0 0 0 46 0 0 0 0 16 16 0 0 16 57
,
 1 1 0 0 72 0 0 0 0 0 0 46 0 0 46 0
,
 72 72 0 0 0 1 0 0 0 0 72 0 0 0 0 1
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,16,16,0,0,16,57],[1,72,0,0,1,0,0,0,0,0,0,46,0,0,46,0],[72,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1] >;

C4⋊C4.150D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4._{150}D_6
% in TeX

G:=Group("C4:C4.150D6");
// GroupNames label

G:=SmallGroup(192,363);
// by ID

G=gap.SmallGroup(192,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,570,136,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b^-1,d*c*d=a^2*c^5>;
// generators/relations

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