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G = C24.27C23  order 192 = 26·3

20th non-split extension by C24 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.24D8, C24.35D4, Q16.7D6, C24.27C23, D24.10C22, Dic12.11C22, (C6xQ16):1C2, (C2xQ16):7S3, C3:Q32:5C2, C6.66(C2xD8), (C2xC8).93D6, (C2xC6).45D8, C4oD24.2C2, C8.6D6:5C2, C3:4(Q32:C2), C8.4(C3:D4), C3:C16.3C22, C12.C8:3C2, C4.18(D4:S3), (C2xC12).183D4, C12.182(C2xD4), C8.33(C22xS3), (C2xC24).33C22, (C3xQ16).7C22, C22.11(D4:S3), C2.21(C2xD4:S3), C4.12(C2xC3:D4), (C2xC4).80(C3:D4), SmallGroup(192,738)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C24.27C23
Chief series
C1 — C3 — C6 — C12 — C24 — D24 — C4oD24 — C24.27C23
Lower central
C3 — C6 — C12 — C24 — C24.27C23
Upper central
C1 — C2 — C2xC4 — C2xC8 — C2xQ16

Generators and relations for C24.27C23
 G = < a,b,c,d | a24=b2=1, c2=d2=a12, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a9b, dcd-1=a12c >

Subgroups: 248 in 82 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, D6, C2xC6, C16, C2xC8, D8, SD16, Q16, Q16, C2xQ8, C4oD4, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xQ8, M5(2), SD32, Q32, C2xQ16, C4oD8, C3:C16, C24:C2, D24, Dic12, C2xC24, C3xQ16, C3xQ16, C4oD12, C6xQ8, Q32:C2, C12.C8, C8.6D6, C3:Q32, C4oD24, C6xQ16, C24.27C23
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C2xD8, D4:S3, C2xC3:D4, Q32:C2, C2xD4:S3, C24.27C23

Character table of C24.27C23

 class 12A2B2C34A4B4C4D4E6A6B6C8A8B8C12A12B12C12D12E12F16A16B16C16D24A24B24C24D
 size 112242228824222224448888121212124444
ρ1111111111111111111111111111111    trivial
ρ211-1-11-111-111-1-111-11-11-1-111-1-11-11-11    linear of order 2
ρ311-111-111-1-11-1-111-11-11-1-11-111-1-11-11    linear of order 2
ρ4111-111111-1111111111111-1-1-1-11111    linear of order 2
ρ5111-1111-1-1-111111111-1-1-1-111111111    linear of order 2
ρ611-111-11-11-11-1-111-11-1-111-11-1-11-11-11    linear of order 2
ρ711-1-11-11-1111-1-111-11-1-111-1-111-1-11-11    linear of order 2
ρ81111111-1-1111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ922-202-220002-2-2-2-222-2000000002-22-2    orthogonal lifted from D4
ρ1022-20-1-222-20-11122-2-11-111-100001-11-1    orthogonal lifted from D6
ρ1122-20-1-22-220-11122-2-111-1-1100001-11-1    orthogonal lifted from D6
ρ122220-122-2-20-1-1-1222-1-111110000-1-1-1-1    orthogonal lifted from D6
ρ132220-122220-1-1-1222-1-1-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ142220222000222-2-2-22200000000-2-2-2-2    orthogonal lifted from D4
ρ1522-2022-20002-2-2000-220000√2√2-√2-√20000    orthogonal lifted from D8
ρ1622-2022-20002-2-2000-220000-√2-√2√2√20000    orthogonal lifted from D8
ρ1722202-2-2000222000-2-20000√2-√2√2-√20000    orthogonal lifted from D8
ρ1822202-2-2000222000-2-20000-√2√2-√2√20000    orthogonal lifted from D8
ρ1922-20-1-22000-111-2-22-11√-3√-3-√-3-√-30000-11-11    complex lifted from C3:D4
ρ202220-122000-1-1-1-2-2-2-1-1√-3-√-3√-3-√-300001111    complex lifted from C3:D4
ρ2122-20-1-22000-111-2-22-11-√-3-√-3√-3√-30000-11-11    complex lifted from C3:D4
ρ222220-122000-1-1-1-2-2-2-1-1-√-3√-3-√-3√-300001111    complex lifted from C3:D4
ρ234440-2-4-4000-2-2-200022000000000000    orthogonal lifted from D4:S3, Schur index 2
ρ2444-40-24-4000-2220002-2000000000000    orthogonal lifted from D4:S3, Schur index 2
ρ254-400400000-400-2√22√20000000000002√20-2√2    symplectic lifted from Q32:C2, Schur index 2
ρ264-400400000-4002√2-2√2000000000000-2√202√2    symplectic lifted from Q32:C2, Schur index 2
ρ274-400-2000002-2√-32√-32√2-2√200000000000√-6√2-√-6-√2    complex faithful
ρ284-400-2000002-2√-32√-3-2√22√200000000000-√-6-√2√-6√2    complex faithful
ρ294-400-20000022√-3-2√-3-2√22√200000000000√-6-√2-√-6√2    complex faithful
ρ304-400-20000022√-3-2√-32√2-2√200000000000-√-6√2√-6-√2    complex faithful

Smallest permutation representation of C24.27C23
►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)

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

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

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

Matrix representation of C24.27C23 ►in GL4(F7) generated by

0222
1053
0455
5366
,
1610
0620
0010
1356
,
2403
0316
3250
3514
,
4301
2052
2265
5264
G:=sub<GL(4,GF(7))| [0,1,0,5,2,0,4,3,2,5,5,6,2,3,5,6],[1,0,0,1,6,6,0,3,1,2,1,5,0,0,0,6],[2,0,3,3,4,3,2,5,0,1,5,1,3,6,0,4],[4,2,2,5,3,0,2,2,0,5,6,6,1,2,5,4] >;
 

C24.27C23 in GAP, Magma, Sage, TeX 

C_{24}._{27}C_2^3
 
% in TeX
 
G:=Group("C24.27C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,738);
 
// by ID
 
G=gap.SmallGroup(192,738);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,184,675,185,192,1684,438,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^24=b^2=1,c^2=d^2=a^12,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=a^12*c>;
 
// generators/relations
 

Export

Character table of C24.27C23 in TeX

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