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G = C42.30D4order 128 = 27

12nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.30D4, C42.11Q8, C23.9C42, C22⋊C8.4C4, (C2×M4(2)).3C4, C4.11(C4.D4), C4⋊M4(2).9C2, C4.11(C4.10D4), C42.6C4.11C2, (C2×C42).139C22, C2.14(M4(2)⋊4C4), C2.10(C22.C42), C22.57(C2.C42), (C2×C4).25(C4⋊C4), (C22×C4).161(C2×C4), (C2×C4).306(C22⋊C4), SmallGroup(128,39)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.30D4
C1C2C22C2×C4C42C2×C42C42.6C4 — C42.30D4
C1C22C23 — C42.30D4
C1C22C2×C42 — C42.30D4
C1C22C22C2×C42 — C42.30D4

Generators and relations for C42.30D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=a-1b-1c3 >

Subgroups: 120 in 65 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, C22×C4 [×2], C8⋊C4 [×2], C22⋊C8 [×4], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C4⋊M4(2), C42.6C4 [×2], C42.30D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C22.C42, M4(2)⋊4C4 [×2], C42.30D4

Character table of C42.30D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111-1-1-1-1-1-11-11-11    linear of order 2
ρ311111111111111-111-1-1-11-1-1-11-1    linear of order 2
ρ411111111111111-1-1-1111-1-11-1-1-1    linear of order 2
ρ511111-11-11-1-1-1-111-i-ii-i-ii-1i-1i1    linear of order 4
ρ6111111-11-1-1-1-11-1-ii-i-1-11-i-i1iii    linear of order 4
ρ711111-11-11-1-1-1-11-1-i-i-iiii1-i1i-1    linear of order 4
ρ811111-1-1-1-1111-1-1-i-11i-ii-1i-i-i1i    linear of order 4
ρ9111111-11-1-1-1-11-1-i-ii11-1i-i-1i-ii    linear of order 4
ρ1011111-1-1-1-1111-1-1i-11-ii-i-1-iii1-i    linear of order 4
ρ1111111-1-1-1-1111-1-1i1-1i-ii1-i-ii-1-i    linear of order 4
ρ1211111-1-1-1-1111-1-1-i1-1-ii-i1ii-i-1i    linear of order 4
ρ13111111-11-1-1-1-11-1ii-i11-1-ii-1-ii-i    linear of order 4
ρ1411111-11-11-1-1-1-11-1iii-i-i-i1i1-i-1    linear of order 4
ρ15111111-11-1-1-1-11-1i-ii-1-11ii1-i-i-i    linear of order 4
ρ1611111-11-11-1-1-1-111ii-iii-i-1-i-1-i1    linear of order 4
ρ172222-22-22-222-2-22000000000000    orthogonal lifted from D4
ρ182222-22222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192222-2-22-2222-22-2000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-2-2-2222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44040-4000000000000000000    orthogonal lifted from C4.D4
ρ224-4-440-404000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ234-44-4000004i-4i000000000000000    complex lifted from M4(2)⋊4C4
ρ2444-4-4004i0-4i00000000000000000    complex lifted from M4(2)⋊4C4
ρ254-44-400000-4i4i000000000000000    complex lifted from M4(2)⋊4C4
ρ2644-4-400-4i04i00000000000000000    complex lifted from M4(2)⋊4C4

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

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

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

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

Matrix representation of C42.30D4 in GL8(𝔽17)

130000000
013000000
001300000
000130000
0000131600
000015400
0000001316
000000154
,
01000000
10000000
00010000
00100000
000013000
000001300
000000130
000000013
,
001550000
001220000
122000000
155000000
0000001212
000000165
000013200
000011400
,
155000000
122000000
001550000
001220000
00000010
00000001
0000131600
000015400

G:=sub<GL(8,GF(17))| [13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[0,0,12,15,0,0,0,0,0,0,2,5,0,0,0,0,15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,0,13,11,0,0,0,0,0,0,2,4,0,0,0,0,12,16,0,0,0,0,0,0,12,5,0,0],[15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,15,12,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,13,15,0,0,0,0,0,0,16,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.30D4 in GAP, Magma, Sage, TeX

C_4^2._{30}D_4
% in TeX

G:=Group("C4^2.30D4");
// GroupNames label

G:=SmallGroup(128,39);
// by ID

G=gap.SmallGroup(128,39);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,248,2804,102]);
// Polycyclic

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

Export

Character table of C42.30D4 in TeX

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