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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, (C6×Q16)⋊1C2, (C2×Q16)⋊7S3, C3⋊Q32⋊5C2, C6.66(C2×D8), (C2×C8).93D6, (C2×C6).45D8, C4○D24.2C2, C8.6D6⋊5C2, C3⋊4(Q32⋊C2), C8.4(C3⋊D4), C3⋊C16.3C22, C12.C8⋊3C2, C4.18(D4⋊S3), (C2×C12).183D4, C12.182(C2×D4), C8.33(C22×S3), (C2×C24).33C22, (C3×Q16).7C22, C22.11(D4⋊S3), C2.21(C2×D4⋊S3), C4.12(C2×C3⋊D4), (C2×C4).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 — C4○D24 — C24.27C23
Lower central
C3 — C6 — C12 — C24 — C24.27C23
Upper central
C1 — C2 — C2×C4 — C2×C8 — C2×Q16

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, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, M5(2), SD32, Q32, C2×Q16, C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C2×C24, C3×Q16, C3×Q16, C4○D12, C6×Q8, Q32⋊C2, C12.C8, C8.6D6, C3⋊Q32, C4○D24, C6×Q16, C24.27C23
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, Q32⋊C2, C2×D4⋊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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