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G = (C2×D12)⋊10C4order 192 = 26·3

6th semidirect product of C2×D12 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C41(D6⋊C4), C6.59(C4×D4), (C2×D12)⋊10C4, C122(C22⋊C4), (C2×Dic3)⋊11D4, (C2×C4).144D12, (C2×C12).142D4, C2.5(C12⋊D4), C2.3(C123D4), C6.15(C41D4), C6.54(C4⋊D4), C22.111(S3×D4), C22.48(C2×D12), (C22×C4).351D6, C6.51(C4.4D4), (C22×D12).11C2, C2.18(Dic35D4), C2.2(C12.23D4), (S3×C23).18C22, C23.306(C22×S3), (C22×C6).352C23, C33(C24.3C22), (C22×C12).144C22, C22.28(Q83S3), (C22×Dic3).192C22, (C6×C4⋊C4)⋊6C2, (C2×C4⋊C4)⋊6S3, (C2×C4×Dic3)⋊1C2, (C2×D6⋊C4)⋊10C2, (C2×C4).79(C4×S3), C2.16(C2×D6⋊C4), (C2×C12).86(C2×C4), (C2×C6).452(C2×D4), C6.43(C2×C22⋊C4), C22.137(S3×C2×C4), C22.67(C2×C3⋊D4), (C2×C6).189(C4○D4), (C2×C4).129(C3⋊D4), (C22×S3).21(C2×C4), (C2×C6).120(C22×C4), SmallGroup(192,547)

Series: Derived Chief Lower central Upper central

C1C2×C6 — (C2×D12)⋊10C4
C1C3C6C2×C6C22×C6S3×C23C22×D12 — (C2×D12)⋊10C4
C3C2×C6 — (C2×D12)⋊10C4
C1C23C2×C4⋊C4

Generators and relations for (C2×D12)⋊10C4
 G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, dcd-1=ac=ca, ad=da, cbc=b-1, dbd-1=b7 >

Subgroups: 824 in 258 conjugacy classes, 83 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×8], C23, C23 [×16], Dic3 [×4], C12 [×4], C12 [×2], D6 [×20], C2×C6 [×3], C2×C6 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×6], C2×C12 [×6], C22×S3 [×4], C22×S3 [×12], C22×C6, C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C4×Dic3 [×2], D6⋊C4 [×8], C3×C4⋊C4 [×2], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23 [×2], C24.3C22, C2×C4×Dic3, C2×D6⋊C4 [×4], C6×C4⋊C4, C22×D12, (C2×D12)⋊10C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, D6⋊C4 [×4], S3×C2×C4, C2×D12, S3×D4 [×2], Q83S3 [×2], C2×C3⋊D4, C24.3C22, Dic35D4 [×2], C12⋊D4 [×2], C2×D6⋊C4, C123D4, C12.23D4, (C2×D12)⋊10C4

Smallest permutation representation of (C2×D12)⋊10C4
On 96 points
Generators in S96
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 71)(14 72)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 93)(26 94)(27 95)(28 96)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(49 83)(50 84)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 72)(23 71)(24 70)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(37 41)(38 40)(42 48)(43 47)(44 46)(49 53)(50 52)(54 60)(55 59)(56 58)(74 84)(75 83)(76 82)(77 81)(78 80)
(1 23 78 28)(2 18 79 35)(3 13 80 30)(4 20 81 25)(5 15 82 32)(6 22 83 27)(7 17 84 34)(8 24 73 29)(9 19 74 36)(10 14 75 31)(11 21 76 26)(12 16 77 33)(37 68 49 95)(38 63 50 90)(39 70 51 85)(40 65 52 92)(41 72 53 87)(42 67 54 94)(43 62 55 89)(44 69 56 96)(45 64 57 91)(46 71 58 86)(47 66 59 93)(48 61 60 88)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I···4P6A···6G12A···12L
order12···222223444444444···46···612···12
size11···1121212122222244446···62···24···4

48 irreducible representations

dim1111112222222244
type++++++++++++
imageC1C2C2C2C2C4S3D4D4D6C4○D4C4×S3D12C3⋊D4S3×D4Q83S3
kernel(C2×D12)⋊10C4C2×C4×Dic3C2×D6⋊C4C6×C4⋊C4C22×D12C2×D12C2×C4⋊C4C2×Dic3C2×C12C22×C4C2×C6C2×C4C2×C4C2×C4C22C22
# reps1141181443444422

Matrix representation of (C2×D12)⋊10C4 in GL6(𝔽13)

100000
010000
0012000
0001200
0000120
0000012
,
010000
12120000
000100
00121200
000005
000050
,
010000
100000
000100
001000
0000120
000001
,
800000
080000
003600
0071000
000001
0000120

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

(C2×D12)⋊10C4 in GAP, Magma, Sage, TeX

(C_2\times D_{12})\rtimes_{10}C_4
% in TeX

G:=Group("(C2xD12):10C4");
// GroupNames label

G:=SmallGroup(192,547);
// by ID

G=gap.SmallGroup(192,547);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,422,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7>;
// generators/relations

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