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G = C24.94D4order 128 = 27

49th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.94D4, C25.33C22, C23.305C24, C24.561C23, C22.1232+ 1+4, (C2×D4).278D4, (C23×C4)⋊9C22, C243C411C2, (D4×C23).11C2, C23.148(C2×D4), C2.13(D45D4), C22.15C22≀C2, C23.10D43C2, C2.7(C233D4), (C22×C4).47C23, C23.8Q822C2, C23.296(C4○D4), C23.23D422C2, C22.185(C22×D4), C2.C4216C22, (C22×D4).498C22, C225(C22.D4), (C2×C4⋊C4)⋊10C22, (C2×C4).301(C2×D4), C2.12(C2×C22≀C2), (C2×C22⋊C4)⋊10C22, (C22×C22⋊C4)⋊11C2, C22.184(C2×C4○D4), (C2×C22.D4)⋊2C2, C2.13(C2×C22.D4), SmallGroup(128,1137)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.94D4
C1C2C22C23C24C23×C4C22×C22⋊C4 — C24.94D4
C1C23 — C24.94D4
C1C23 — C24.94D4
C1C23 — C24.94D4

Generators and relations for C24.94D4
 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=ce-1 >

Subgroups: 1236 in 558 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×14], C4 [×12], C22, C22 [×14], C22 [×86], C2×C4 [×4], C2×C4 [×44], D4 [×32], C23, C23 [×18], C23 [×94], C22⋊C4 [×30], C4⋊C4 [×8], C22×C4 [×10], C22×C4 [×16], C2×D4 [×8], C2×D4 [×52], C24 [×3], C24 [×6], C24 [×22], C2.C42 [×4], C2×C22⋊C4 [×18], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C22.D4 [×8], C23×C4 [×3], C22×D4 [×4], C22×D4 [×12], C25 [×2], C243C4, C243C4 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.10D4 [×4], C22×C22⋊C4, C2×C22.D4 [×2], D4×C23, C24.94D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C22.D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C22≀C2, C2×C22.D4, C233D4, D45D4 [×4], C24.94D4

Smallest permutation representation of C24.94D4
On 32 points
Generators in S32
(1 19)(2 29)(3 17)(4 31)(5 7)(6 12)(8 10)(9 11)(13 23)(14 16)(15 21)(18 27)(20 25)(22 24)(26 30)(28 32)
(1 26)(2 29)(3 28)(4 31)(5 16)(6 12)(7 14)(8 10)(9 24)(11 22)(13 23)(15 21)(17 32)(18 27)(19 30)(20 25)
(1 17)(2 18)(3 19)(4 20)(5 22)(6 23)(7 24)(8 21)(9 14)(10 15)(11 16)(12 13)(25 31)(26 32)(27 29)(28 30)
(1 30)(2 31)(3 32)(4 29)(5 14)(6 15)(7 16)(8 13)(9 22)(10 23)(11 24)(12 21)(17 28)(18 25)(19 26)(20 27)
(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 14)(2 12)(3 16)(4 10)(5 30)(6 27)(7 32)(8 25)(9 17)(11 19)(13 18)(15 20)(21 31)(22 28)(23 29)(24 26)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O2P···2U4A···4L4M4N4O4P
order12···22···22···24···44444
size11···12···24···44···48888

38 irreducible representations

dim111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC24.94D4C243C4C23.8Q8C23.23D4C23.10D4C22×C22⋊C4C2×C22.D4D4×C23C2×D4C24C23C22
# reps132241218482

Matrix representation of C24.94D4 in GL6(𝔽5)

400000
040000
001000
000400
000040
000011
,
400000
040000
004000
000400
000040
000011
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
400000
000100
001000
000031
000022
,
400000
010000
001000
000100
000043
000001

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

C24.94D4 in GAP, Magma, Sage, TeX

C_2^4._{94}D_4
% in TeX

G:=Group("C2^4.94D4");
// GroupNames label

G:=SmallGroup(128,1137);
// by ID

G=gap.SmallGroup(128,1137);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,675]);
// 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=c*e^-1>;
// generators/relations

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