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G = C24.15D4order 128 = 27

15th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.15D4, C23⋊C87C2, C4⋊C4.15D4, (C2×D4).18D4, (C2×Q8).18D4, (C22×C4).17D4, C23.529(C2×D4), C22.D83C2, C2.11(D4⋊D4), C22.25(C4○D8), C23.11D42C2, C4⋊D4.12C22, C23.31D48C2, C22.SD1614C2, (C22×C4).18C23, C22⋊Q8.12C22, C2.11(D4.9D4), C22.139C22≀C2, C23.47D427C2, C22⋊C8.117C22, C22.21(C8⋊C22), C22.32C24.2C2, C2.9(C23.7D4), C2.C42.25C22, (C2×C4).207(C2×D4), (C2×C4⋊C4).23C22, (C2×C22⋊C4).98C22, SmallGroup(128,344)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.15D4
C1C2C22C23C22×C4C2×C22⋊C4C22.32C24 — C24.15D4
C1C22C22×C4 — C24.15D4
C1C22C22×C4 — C24.15D4
C1C2C22C22×C4 — C24.15D4

Generators and relations for C24.15D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=d, eae-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 308 in 116 conjugacy classes, 32 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C22 [×3], C22 [×11], C8 [×2], C2×C4 [×2], C2×C4 [×13], D4 [×5], Q8, C23, C23 [×6], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C24, C2.C42, C2.C42, C22⋊C8 [×2], D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C23⋊C8, C22.SD16, C23.31D4, C23.11D4, C22.D8, C23.47D4, C22.32C24, C24.15D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8, C8⋊C22, D4⋊D4, D4.9D4, C23.7D4, C24.15D4

Character table of C24.15D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288444488888888888
ρ111111111111111111111111    trivial
ρ21111111-11-1-11-11-11-11-1-11-11    linear of order 2
ρ3111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ41111111-11-1-1111-1-1-1-111-11-1    linear of order 2
ρ5111111-1-11-1-11-1-11111-11-11-1    linear of order 2
ρ6111111-1111111-1-11-111-1-1-1-1    linear of order 2
ρ7111111-1-11-1-111-11-11-11-11-11    linear of order 2
ρ8111111-111111-1-1-1-1-1-1-11111    linear of order 2
ρ92222-2-2-20-200202000000000    orthogonal lifted from D4
ρ102222-2-200200-200-202000000    orthogonal lifted from D4
ρ112222220-2-222-200000000000    orthogonal lifted from D4
ρ122222-2-200200-20020-2000000    orthogonal lifted from D4
ρ1322222202-2-2-2-200000000000    orthogonal lifted from D4
ρ142222-2-220-20020-2000000000    orthogonal lifted from D4
ρ1522-2-2-22000-2i2i00000000-2-2--22    complex lifted from C4○D8
ρ1622-2-2-22000-2i2i00000000--22-2-2    complex lifted from C4○D8
ρ1722-2-2-220002i-2i00000000-22--2-2    complex lifted from C4○D8
ρ1822-2-2-220002i-2i00000000--2-2-22    complex lifted from C4○D8
ρ1944-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4400000000-2i000002i0000    complex lifted from D4.9D4
ρ214-4-44000000002i00000-2i0000    complex lifted from D4.9D4
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

Smallest permutation representation of C24.15D4
On 32 points
Generators in S32
(3 31)(4 32)(7 27)(8 28)(9 13)(10 14)(11 24)(12 17)(15 20)(16 21)(18 22)(19 23)
(2 30)(4 32)(6 26)(8 28)(10 19)(12 21)(14 23)(16 17)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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)
(1 13 29 22)(2 21 30 12)(3 11 31 20)(4 19 32 10)(5 9 25 18)(6 17 26 16)(7 15 27 24)(8 23 28 14)

G:=sub<Sym(32)| (3,31)(4,32)(7,27)(8,28)(9,13)(10,14)(11,24)(12,17)(15,20)(16,21)(18,22)(19,23), (2,30)(4,32)(6,26)(8,28)(10,19)(12,21)(14,23)(16,17), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,13,29,22)(2,21,30,12)(3,11,31,20)(4,19,32,10)(5,9,25,18)(6,17,26,16)(7,15,27,24)(8,23,28,14)>;

G:=Group( (3,31)(4,32)(7,27)(8,28)(9,13)(10,14)(11,24)(12,17)(15,20)(16,21)(18,22)(19,23), (2,30)(4,32)(6,26)(8,28)(10,19)(12,21)(14,23)(16,17), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,13,29,22)(2,21,30,12)(3,11,31,20)(4,19,32,10)(5,9,25,18)(6,17,26,16)(7,15,27,24)(8,23,28,14) );

G=PermutationGroup([(3,31),(4,32),(7,27),(8,28),(9,13),(10,14),(11,24),(12,17),(15,20),(16,21),(18,22),(19,23)], [(2,30),(4,32),(6,26),(8,28),(10,19),(12,21),(14,23),(16,17)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(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)], [(1,13,29,22),(2,21,30,12),(3,11,31,20),(4,19,32,10),(5,9,25,18),(6,17,26,16),(7,15,27,24),(8,23,28,14)])

Matrix representation of C24.15D4 in GL6(𝔽17)

100000
1160000
001001
0001600
0000116
0000016
,
100000
010000
001001
000101
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
900000
1220000
0001300
00013130
0013400
000804
,
2130000
5150000
00130130
00013130
000040
0000813

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,1,0,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,1,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,12,0,0,0,0,0,2,0,0,0,0,0,0,0,0,13,0,0,0,13,13,4,8,0,0,0,13,0,0,0,0,0,0,0,4],[2,5,0,0,0,0,13,15,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,13,13,4,8,0,0,0,0,0,13] >;

C24.15D4 in GAP, Magma, Sage, TeX

C_2^4._{15}D_4
% in TeX

G:=Group("C2^4.15D4");
// GroupNames label

G:=SmallGroup(128,344);
// by ID

G=gap.SmallGroup(128,344);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,520,1123,570,521,136,1411]);
// Polycyclic

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

Export

Character table of C24.15D4 in TeX

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