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## G = C24.23D6order 192 = 26·3

### 12nd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.23D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.23D6
 Lower central C3 — C2×C6 — C24.23D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.23D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 808 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C23.23D4, C6.C42, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C22×C4, C22×C3⋊D4, C24.23D6
Quotients:

Smallest permutation representation of C24.23D6
On 96 points
Generators in S96
(1 76)(2 70)(3 78)(4 72)(5 80)(6 62)(7 82)(8 64)(9 84)(10 66)(11 74)(12 68)(13 89)(14 43)(15 91)(16 45)(17 93)(18 47)(19 95)(20 37)(21 85)(22 39)(23 87)(24 41)(25 79)(26 61)(27 81)(28 63)(29 83)(30 65)(31 73)(32 67)(33 75)(34 69)(35 77)(36 71)(38 53)(40 55)(42 57)(44 59)(46 49)(48 51)(50 94)(52 96)(54 86)(56 88)(58 90)(60 92)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 61)(21 62)(22 63)(23 64)(24 65)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 73)(58 74)(59 75)(60 76)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 71 19 65)(14 64 20 70)(15 69 21 63)(16 62 22 68)(17 67 23 61)(18 72 24 66)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 46 43 40)(38 39 44 45)(41 42 47 48)(49 74 55 80)(50 79 56 73)(51 84 57 78)(52 77 58 83)(53 82 59 76)(54 75 60 81)(85 86 91 92)(87 96 93 90)(88 89 94 95)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.23D6 C6.C42 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 S3×C22×C4 C22×C3⋊D4 C2×C3⋊D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 1 1 1 8 1 2 2 4 2 1 4 4 4 4 3 1

Matrix representation of C24.23D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 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 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 0 0 0 12 0 0 0 0 0 0 12
,
 5 5 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 5 5 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,8,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C24.23D6 in GAP, Magma, Sage, TeX

C_2^4._{23}D_6
% in TeX

G:=Group("C2^4.23D6");
// GroupNames label

G:=SmallGroup(192,515);
// by ID

G=gap.SmallGroup(192,515);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,6278]);
// Polycyclic

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

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