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G = C24.2Q8order 128 = 27

1st non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.2Q8, C22⋊C8.3C4, (C22×C8).4C4, (C2×C4).37C42, C23.41(C4⋊C4), C4.23(C23⋊C4), (C22×C4).181D4, C24.4C4.5C2, C22.2(C8.C4), C2.5(C4.C42), (C23×C4).191C22, C2.5(C23.9D4), C2.4(C4.10C42), C22.43(C2.C42), (C2×C22⋊C8).4C2, (C22×C4).93(C2×C4), (C2×C4).298(C22⋊C4), SmallGroup(128,25)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.2Q8
C1C2C22C2×C4C22×C4C23×C4C2×C22⋊C8 — C24.2Q8
C1C2C2×C4 — C24.2Q8
C1C22C23×C4 — C24.2Q8
C1C2C22C23×C4 — C24.2Q8

Generators and relations for C24.2Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=dc=cd, f2=bcde2, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf-1=bcd, ce=ec, cf=fc, de=ed, df=fd, fef-1=ace3 >

Subgroups: 200 in 92 conjugacy classes, 34 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×12], C8 [×6], C2×C4 [×2], C2×C4 [×2], C2×C4 [×9], C23, C23 [×2], C23 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×2], C24, C22⋊C8 [×4], C22⋊C8 [×4], C22×C8 [×2], C2×M4(2) [×2], C23×C4, C2×C22⋊C8, C24.4C4 [×2], C24.2Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C8.C4 [×2], C4.10C42, C4.C42, C23.9D4, C24.2Q8

Smallest permutation representation of C24.2Q8
On 32 points
Generators in S32
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)
(2 18)(4 20)(6 22)(8 24)(9 13)(10 31)(11 15)(12 25)(14 27)(16 29)(26 30)(28 32)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(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 28 7 30 5 32 3 26)(2 27 24 12 6 31 20 16)(4 25 18 10 8 29 22 14)(9 17 11 23 13 21 15 19)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H8A···8H8I···8P
order122222224···48···88···8
size111122442···24···48···8

32 irreducible representations

dim1111122244
type++++-+
imageC1C2C2C4C4D4Q8C8.C4C23⋊C4C4.10C42
kernelC24.2Q8C2×C22⋊C8C24.4C4C22⋊C8C22×C8C22×C4C24C22C4C2
# reps1128431822

Matrix representation of C24.2Q8 in GL6(𝔽17)

1600000
0160000
0016000
0001600
000010
000001
,
1600000
010000
001000
0001600
000010
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
900000
0150000
000100
001000
0000016
0000160
,
010000
1300000
000010
000001
001000
0001600

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

C24.2Q8 in GAP, Magma, Sage, TeX

C_2^4._2Q_8
% in TeX

G:=Group("C2^4.2Q8");
// GroupNames label

G:=SmallGroup(128,25);
// by ID

G=gap.SmallGroup(128,25);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,360,3924,242]);
// Polycyclic

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

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