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G = C42.292D4order 128 = 27

274th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.292D4, C42.422C23, C4.642- 1+4, C8⋊Q818C2, D4.Q824C2, Q8.Q824C2, C4⋊C8.74C22, (C2×C8).74C23, C4⋊C4.179C23, (C2×C4).438C24, C23.301(C2×D4), (C22×C4).520D4, C4⋊Q8.321C22, C8⋊C4.31C22, C42.6C420C2, C4.Q8.42C22, (C4×D4).120C22, (C2×D4).182C23, C22⋊C8.65C22, (C2×Q8).170C23, (C4×Q8).117C22, C2.D8.108C22, D4⋊C4.52C22, C4⋊D4.205C22, C23.20D428C2, (C2×C42).899C22, Q8⋊C4.52C22, C23.19D4.3C2, C22.698(C22×D4), C22⋊Q8.210C22, C2.67(D8⋊C22), (C22×C4).1103C23, C42.28C229C2, C4.4D4.162C22, C42.C2.139C22, C42⋊C2.168C22, C23.37C2323C2, C23.36C23.29C2, C2.86(C23.38C23), (C2×C4).562(C2×D4), SmallGroup(128,1972)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.292D4
C1C2C4C2×C4C22×C4C42⋊C2C23.37C23 — C42.292D4
C1C2C2×C4 — C42.292D4
C1C22C2×C42 — C42.292D4
C1C2C2C2×C4 — C42.292D4

Generators and relations for C42.292D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >

Subgroups: 292 in 164 conjugacy classes, 84 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×4], Q8 [×6], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C42.C2 [×2], C422C2, C4⋊Q8 [×2], C42.6C4, D4.Q8 [×2], Q8.Q8 [×2], C23.19D4 [×2], C23.20D4 [×2], C42.28C22 [×2], C8⋊Q8 [×2], C23.36C23, C23.37C23, C42.292D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- 1+4 [×2], C23.38C23, D8⋊C22 [×2], C42.292D4

Character table of C42.292D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11114822222244488888888888
ρ111111111111111111111111111    trivial
ρ21111-11-11-11-1-1-111-1-1-111-111-11-1    linear of order 2
ρ311111-11111111111-1111-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-11-11-1-1-111-11-1111-1-11-11    linear of order 2
ρ51111-1111-111-11-1-111-1-11-1-11-1-11    linear of order 2
ρ6111111-1111-11-1-1-1-1-11-111-111-1-1    linear of order 2
ρ71111-1-111-111-11-1-11-1-1-1111-111-1    linear of order 2
ρ811111-1-1111-11-1-1-1-111-11-11-1-111    linear of order 2
ρ91111-1-1-11-11-1-1-111111-1-11-11-11-1    linear of order 2
ρ1011111-1111111111-1-1-1-1-1-1-11111    linear of order 2
ρ111111-11-11-11-1-1-1111-11-1-1-11-11-11    linear of order 2
ρ12111111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ1311111-1-1111-11-1-1-111-11-1-1111-1-1    linear of order 2
ρ141111-1-111-111-11-1-1-1-111-1111-1-11    linear of order 2
ρ15111111-1111-11-1-1-11-1-11-11-1-1-111    linear of order 2
ρ161111-1111-111-11-1-1-1111-1-1-1-111-1    linear of order 2
ρ17222220-2-2-2-2-2-22-2200000000000    orthogonal lifted from D4
ρ182222-202-22-222-2-2200000000000    orthogonal lifted from D4
ρ192222-20-2-22-2-2222-200000000000    orthogonal lifted from D4
ρ202222202-2-2-22-2-22-200000000000    orthogonal lifted from D4
ρ214-44-4000-4040000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-44-400040-40000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ234-4-440000-4i004i00000000000000    complex lifted from D8⋊C22
ρ2444-4-4004i000-4i000000000000000    complex lifted from D8⋊C22
ρ2544-4-400-4i0004i000000000000000    complex lifted from D8⋊C22
ρ264-4-4400004i00-4i00000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.292D4
On 64 points
Generators in S64
(1 61 25 50)(2 58 26 55)(3 63 27 52)(4 60 28 49)(5 57 29 54)(6 62 30 51)(7 59 31 56)(8 64 32 53)(9 18 38 47)(10 23 39 44)(11 20 40 41)(12 17 33 46)(13 22 34 43)(14 19 35 48)(15 24 36 45)(16 21 37 42)
(1 38 5 34)(2 14 6 10)(3 40 7 36)(4 16 8 12)(9 29 13 25)(11 31 15 27)(17 60 21 64)(18 54 22 50)(19 62 23 58)(20 56 24 52)(26 35 30 39)(28 37 32 33)(41 59 45 63)(42 53 46 49)(43 61 47 57)(44 55 48 51)
(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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 34)(10 16)(11 40)(12 14)(13 38)(15 36)(17 44)(19 42)(20 24)(21 48)(23 46)(27 31)(33 35)(37 39)(41 45)(49 51)(50 57)(52 63)(53 55)(54 61)(56 59)(58 64)(60 62)

G:=sub<Sym(64)| (1,61,25,50)(2,58,26,55)(3,63,27,52)(4,60,28,49)(5,57,29,54)(6,62,30,51)(7,59,31,56)(8,64,32,53)(9,18,38,47)(10,23,39,44)(11,20,40,41)(12,17,33,46)(13,22,34,43)(14,19,35,48)(15,24,36,45)(16,21,37,42), (1,38,5,34)(2,14,6,10)(3,40,7,36)(4,16,8,12)(9,29,13,25)(11,31,15,27)(17,60,21,64)(18,54,22,50)(19,62,23,58)(20,56,24,52)(26,35,30,39)(28,37,32,33)(41,59,45,63)(42,53,46,49)(43,61,47,57)(44,55,48,51), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,34)(10,16)(11,40)(12,14)(13,38)(15,36)(17,44)(19,42)(20,24)(21,48)(23,46)(27,31)(33,35)(37,39)(41,45)(49,51)(50,57)(52,63)(53,55)(54,61)(56,59)(58,64)(60,62)>;

G:=Group( (1,61,25,50)(2,58,26,55)(3,63,27,52)(4,60,28,49)(5,57,29,54)(6,62,30,51)(7,59,31,56)(8,64,32,53)(9,18,38,47)(10,23,39,44)(11,20,40,41)(12,17,33,46)(13,22,34,43)(14,19,35,48)(15,24,36,45)(16,21,37,42), (1,38,5,34)(2,14,6,10)(3,40,7,36)(4,16,8,12)(9,29,13,25)(11,31,15,27)(17,60,21,64)(18,54,22,50)(19,62,23,58)(20,56,24,52)(26,35,30,39)(28,37,32,33)(41,59,45,63)(42,53,46,49)(43,61,47,57)(44,55,48,51), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,34)(10,16)(11,40)(12,14)(13,38)(15,36)(17,44)(19,42)(20,24)(21,48)(23,46)(27,31)(33,35)(37,39)(41,45)(49,51)(50,57)(52,63)(53,55)(54,61)(56,59)(58,64)(60,62) );

G=PermutationGroup([(1,61,25,50),(2,58,26,55),(3,63,27,52),(4,60,28,49),(5,57,29,54),(6,62,30,51),(7,59,31,56),(8,64,32,53),(9,18,38,47),(10,23,39,44),(11,20,40,41),(12,17,33,46),(13,22,34,43),(14,19,35,48),(15,24,36,45),(16,21,37,42)], [(1,38,5,34),(2,14,6,10),(3,40,7,36),(4,16,8,12),(9,29,13,25),(11,31,15,27),(17,60,21,64),(18,54,22,50),(19,62,23,58),(20,56,24,52),(26,35,30,39),(28,37,32,33),(41,59,45,63),(42,53,46,49),(43,61,47,57),(44,55,48,51)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,34),(10,16),(11,40),(12,14),(13,38),(15,36),(17,44),(19,42),(20,24),(21,48),(23,46),(27,31),(33,35),(37,39),(41,45),(49,51),(50,57),(52,63),(53,55),(54,61),(56,59),(58,64),(60,62)])

Matrix representation of C42.292D4 in GL8(𝔽17)

99010000
131316160000
991200000
56900000
00000040
00000004
000013000
000001300
,
40000000
04000000
00400000
00040000
00000010
00000001
000016000
000001600
,
91616150000
00100000
10000000
6131280000
00004131212
000044512
00001212134
00005121313
,
10000000
016000000
91616150000
01010000
00001000
000001600
000000160
00000001

G:=sub<GL(8,GF(17))| [9,13,9,5,0,0,0,0,9,13,9,6,0,0,0,0,0,16,12,9,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[9,0,1,6,0,0,0,0,16,0,0,13,0,0,0,0,16,1,0,12,0,0,0,0,15,0,0,8,0,0,0,0,0,0,0,0,4,4,12,5,0,0,0,0,13,4,12,12,0,0,0,0,12,5,13,13,0,0,0,0,12,12,4,13],[1,0,9,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;

C42.292D4 in GAP, Magma, Sage, TeX

C_4^2._{292}D_4
% in TeX

G:=Group("C4^2.292D4");
// GroupNames label

G:=SmallGroup(128,1972);
// by ID

G=gap.SmallGroup(128,1972);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,100,675,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations

Export

Character table of C42.292D4 in TeX

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