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G = C24.7Q8order 128 = 27

6th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.7Q8, (C2×C4).21C42, C4.20(C2×C42), (C22×C4).37Q8, C23.76(C2×Q8), C23.23(C4⋊C4), (C22×C4).256D4, C4.C4216C2, M4(2).28(C2×C4), (C2×M4(2)).26C4, (C23×C4).219C22, (C22×C8).373C22, C2.2(M4(2).C4), C4.20(C2.C42), (C22×C4).1304C23, (C22×M4(2)).14C2, (C2×M4(2)).298C22, C22.12(C2.C42), (C2×C4).40(C4⋊C4), (C2×C8).127(C2×C4), C22.11(C2×C4⋊C4), C4.83(C2×C22⋊C4), (C2×C4).1295(C2×D4), (C2×C4).518(C22×C4), (C22×C4).255(C2×C4), (C2×C4).253(C22⋊C4), C2.15(C2×C2.C42), SmallGroup(128,470)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C24.7Q8
C1C2C4C2×C4C22×C4C23×C4C22×M4(2) — C24.7Q8
C1C2C4 — C24.7Q8
C1C2×C4C23×C4 — C24.7Q8
C1C2C2C22×C4 — C24.7Q8

Generators and relations for C24.7Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=cde2, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=bde3 >

Subgroups: 276 in 186 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×10], C8 [×12], C2×C4 [×2], C2×C4 [×26], C23, C23 [×6], C23 [×2], C2×C8 [×4], C2×C8 [×16], M4(2) [×8], M4(2) [×20], C22×C4 [×2], C22×C4 [×12], C24, C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×16], C2×M4(2) [×10], C23×C4, C4.C42 [×4], C22×M4(2), C22×M4(2) [×2], C24.7Q8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, M4(2).C4 [×2], C24.7Q8

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8X
order12222···244444···48···8
size11112···211112···24···4

44 irreducible representations

dim11112224
type++++--
imageC1C2C2C4D4Q8Q8M4(2).C4
kernelC24.7Q8C4.C42C22×M4(2)C2×M4(2)C22×C4C22×C4C24C2
# reps143246114

Matrix representation of C24.7Q8 in GL6(𝔽17)

100000
010000
001000
00131600
000010
0010016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
0040160
0010016
,
100000
010000
0016000
0001600
0000160
0000016
,
010000
100000
00131500
006400
0071304
00016160
,
530000
14120000
0040150
000041
00100130
001113160

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,4,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,6,7,0,0,0,15,4,13,16,0,0,0,0,0,16,0,0,0,0,4,0],[5,14,0,0,0,0,3,12,0,0,0,0,0,0,4,0,10,11,0,0,0,0,0,13,0,0,15,4,13,16,0,0,0,1,0,0] >;

C24.7Q8 in GAP, Magma, Sage, TeX

C_2^4._7Q_8
% in TeX

G:=Group("C2^4.7Q8");
// GroupNames label

G:=SmallGroup(128,470);
// by ID

G=gap.SmallGroup(128,470);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,172,4037,124]);
// Polycyclic

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

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