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G = C24.28D4order 192 = 26·3

28th non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.28D4, C3⋊C8.22D4, (C6×Q16)⋊6C2, (C2×Q16)⋊6S3, C4.29(S3×D4), (C8×Dic3)⋊7C2, (C2×C8).244D6, (C2×Q8).91D6, (C2×D24).11C2, C6.82(C4○D8), C12.189(C2×D4), C8.19(C3⋊D4), C35(C8.12D4), C12.23D46C2, C6.35(C41D4), (C2×C24).96C22, C22.282(S3×D4), (C6×Q8).94C22, C2.26(C123D4), (C2×C12).465C23, (C2×Dic3).118D4, C2.19(D24⋊C2), (C2×D12).127C22, (C4×Dic3).245C22, C4.16(C2×C3⋊D4), (C2×C6).376(C2×D4), (C2×Q82S3)⋊21C2, (C2×C3⋊C8).279C22, (C2×C4).553(C22×S3), SmallGroup(192,750)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.28D4
C1C3C6C2×C6C2×C12C2×D12C2×D24 — C24.28D4
C3C6C2×C12 — C24.28D4
C1C22C2×C4C2×Q16

Generators and relations for C24.28D4
 G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a-1, cbc=a12b-1 >

Subgroups: 440 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C3⋊C8 [×2], C24 [×2], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3 [×2], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, D24 [×2], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×4], Q82S3 [×4], C2×C24, C3×Q16 [×2], C2×D12 [×2], C6×Q8 [×2], C8.12D4, C8×Dic3, C2×D24, C2×Q82S3 [×2], C12.23D4 [×2], C6×Q16, C24.28D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C4○D8 [×2], S3×D4 [×2], C2×C3⋊D4, C8.12D4, D24⋊C2 [×2], C123D4, C24.28D4

Smallest permutation representation of C24.28D4
On 96 points
Generators in S96
(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 49 91 33)(2 66 92 26)(3 59 93 43)(4 52 94 36)(5 69 95 29)(6 62 96 46)(7 55 73 39)(8 72 74 32)(9 65 75 25)(10 58 76 42)(11 51 77 35)(12 68 78 28)(13 61 79 45)(14 54 80 38)(15 71 81 31)(16 64 82 48)(17 57 83 41)(18 50 84 34)(19 67 85 27)(20 60 86 44)(21 53 87 37)(22 70 88 30)(23 63 89 47)(24 56 90 40)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 72)(47 71)(48 70)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(86 96)(87 95)(88 94)(89 93)(90 92)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A24B24C24D
order1222223444444446668888888812121212121224242424
size11112424222666688222222266664488884444

36 irreducible representations

dim11111122222222444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6C3⋊D4C4○D8S3×D4S3×D4D24⋊C2
kernelC24.28D4C8×Dic3C2×D24C2×Q82S3C12.23D4C6×Q16C2×Q16C3⋊C8C24C2×Dic3C2×C8C2×Q8C8C6C4C22C2
# reps11122112221248114

Matrix representation of C24.28D4 in GL6(𝔽73)

1130000
8620000
00727200
001000
00001657
00001616
,
62700000
65110000
001000
00727200
0000270
0000027
,
100000
17720000
001000
00727200
000010
0000072

G:=sub<GL(6,GF(73))| [11,8,0,0,0,0,3,62,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[62,65,0,0,0,0,70,11,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[1,17,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;

C24.28D4 in GAP, Magma, Sage, TeX

C_{24}._{28}D_4
% in TeX

G:=Group("C24.28D4");
// GroupNames label

G:=SmallGroup(192,750);
// by ID

G=gap.SmallGroup(192,750);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^12*b^-1>;
// generators/relations

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