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G = C247D4order 128 = 27

2nd semidirect product of C24 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C247D4, C251C22, C23.303C24, C24.560C23, C22.1212+ 1+4, C2.6D42, (C2×D4)⋊40D4, C235(C2×D4), (D4×C23)⋊2C2, C222C22≀C2, C232D43C2, C243C410C2, (C23×C4)⋊18C22, (C22×D4)⋊2C22, C2.6(C233D4), C23.23D420C2, (C22×C4).785C23, C22.183(C22×D4), C2.C4214C22, (C2×C4)⋊8(C2×D4), (C2×C22≀C2)⋊2C2, C2.10(C2×C22≀C2), (C2×C22⋊C4)⋊8C22, SmallGroup(128,1135)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C247D4
C1C2C22C23C24C25D4×C23 — C247D4
C1C23 — C247D4
C1C23 — C247D4
C1C23 — C247D4

Generators and relations for C247D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1780 in 782 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×18], C4 [×12], C22, C22 [×14], C22 [×122], C2×C4 [×8], C2×C4 [×36], D4 [×80], C23, C23 [×22], C23 [×138], C22⋊C4 [×26], C22×C4 [×8], C22×C4 [×12], C2×D4 [×16], C2×D4 [×124], C24, C24 [×10], C24 [×34], C2.C42 [×4], C2×C22⋊C4 [×14], C22≀C2 [×16], C23×C4 [×2], C22×D4 [×12], C22×D4 [×24], C25, C25 [×2], C243C4, C23.23D4 [×4], C232D4 [×4], C2×C22≀C2 [×4], D4×C23 [×2], C247D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×20], C23 [×15], C2×D4 [×30], C24, C22≀C2 [×8], C22×D4 [×5], 2+ 1+4 [×2], C2×C22≀C2 [×2], C233D4, D42 [×4], C247D4

Smallest permutation representation of C247D4
On 32 points
Generators in S32
(1 29)(2 12)(3 31)(4 10)(5 18)(6 23)(7 20)(8 21)(9 16)(11 14)(13 32)(15 30)(17 28)(19 26)(22 25)(24 27)
(1 27)(2 13)(3 25)(4 15)(5 16)(6 28)(7 14)(8 26)(9 18)(10 30)(11 20)(12 32)(17 23)(19 21)(22 31)(24 29)
(1 5)(2 6)(3 7)(4 8)(9 24)(10 21)(11 22)(12 23)(13 28)(14 25)(15 26)(16 27)(17 32)(18 29)(19 30)(20 31)
(1 25)(2 26)(3 27)(4 28)(5 14)(6 15)(7 16)(8 13)(9 20)(10 17)(11 18)(12 19)(21 32)(22 29)(23 30)(24 31)
(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 2)(3 4)(5 6)(7 8)(9 17)(10 20)(11 19)(12 18)(13 16)(14 15)(21 31)(22 30)(23 29)(24 32)(25 26)(27 28)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O2P···2Y4A···4H4I4J4K4L
order12···22···22···24···44444
size11···12···24···44···48888

38 irreducible representations

dim111111224
type+++++++++
imageC1C2C2C2C2C2D4D42+ 1+4
kernelC247D4C243C4C23.23D4C232D4C2×C22≀C2D4×C23C2×D4C24C22
# reps1144421642

Matrix representation of C247D4 in GL6(𝔽5)

010000
100000
001000
000100
000010
000004
,
100000
010000
004000
000400
000040
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
400000
003200
000200
000004
000010
,
040000
400000
002300
004300
000001
000010

G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C247D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_7D_4
% in TeX

G:=Group("C2^4:7D4");
// GroupNames label

G:=SmallGroup(128,1135);
// by ID

G=gap.SmallGroup(128,1135);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675]);
// Polycyclic

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

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