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G = C24.70D4order 128 = 27

25th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.70D4, C22.27C4≀C2, C42⋊C219C4, C426C414C2, (C22×C4).679D4, C23.551(C2×D4), C4.21(C42⋊C2), C23.77(C22⋊C4), C24.4C4.20C2, (C23×C4).246C22, (C2×C42).252C22, (C22×C4).1337C23, C42⋊C2.268C22, C2.40(C42⋊C22), C4.130(C22.D4), C2.14(C23.34D4), (C2×M4(2)).162C22, C22.44(C22.D4), (C2×C4⋊C4)⋊30C4, C2.40(C2×C4≀C2), C4⋊C4.197(C2×C4), (C2×C4).1518(C2×D4), (C4×C22⋊C4).12C2, (C2×C4).743(C4○D4), (C22×C4).269(C2×C4), (C2×C4).370(C22×C4), (C2×C4).128(C22⋊C4), (C2×C42⋊C2).18C2, C22.260(C2×C22⋊C4), SmallGroup(128,558)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.70D4
C1C2C4C2×C4C22×C4C42⋊C2C2×C42⋊C2 — C24.70D4
C1C2C2×C4 — C24.70D4
C1C2×C4C23×C4 — C24.70D4
C1C2C2C22×C4 — C24.70D4

Generators and relations for C24.70D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bcd, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 300 in 153 conjugacy classes, 54 normal (26 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×2], C2×C4 [×4], C2×C4 [×4], C2×C4 [×26], C23 [×3], C23 [×5], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×6], C22×C4 [×8], C24, C2.C42, C22⋊C8 [×2], C2×C42 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C2×M4(2) [×2], C23×C4, C426C4 [×4], C4×C22⋊C4, C24.4C4, C2×C42⋊C2, C24.70D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C4≀C2 [×2], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, C2×C4≀C2, C42⋊C22, C24.70D4

Smallest permutation representation of C24.70D4
On 32 points
Generators in S32
(1 9)(2 23)(3 11)(4 17)(5 13)(6 19)(7 15)(8 21)(10 31)(12 25)(14 27)(16 29)(18 26)(20 28)(22 30)(24 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(3 7)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)
(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 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 9 30 22)(2 12 27 21)(3 11 32 24)(4 14 29 23)(5 13 26 18)(6 16 31 17)(7 15 28 20)(8 10 25 19)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I···4Y8A8B8C8D
order122222222444444444···48888
size111122224111122224···48888

38 irreducible representations

dim111111122224
type+++++++
imageC1C2C2C2C2C4C4D4D4C4○D4C4≀C2C42⋊C22
kernelC24.70D4C426C4C4×C22⋊C4C24.4C4C2×C42⋊C2C2×C4⋊C4C42⋊C2C22×C4C24C2×C4C22C2
# reps141114431882

Matrix representation of C24.70D4 in GL4(𝔽17) generated by

16000
01600
00013
0040
,
1000
0100
00160
00016
,
16000
0100
0010
0001
,
16000
01600
0010
0001
,
01300
16000
00130
0004
,
16000
01300
0004
0040
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,4,0,0,13,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,13,0,0,0,0,0,13,0,0,0,0,4],[16,0,0,0,0,13,0,0,0,0,0,4,0,0,4,0] >;

C24.70D4 in GAP, Magma, Sage, TeX

C_2^4._{70}D_4
% in TeX

G:=Group("C2^4.70D4");
// GroupNames label

G:=SmallGroup(128,558);
// by ID

G=gap.SmallGroup(128,558);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,718,172,124]);
// Polycyclic

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

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