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G = C24.10Q8order 128 = 27

9th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.10Q8, M4(2).39D4, C4.139(C4×D4), C22⋊C8.14C4, C4.116C22≀C2, C23.85(C2×Q8), (C22×C4).51Q8, C23.30(C4⋊C4), C4.C421C2, (C22×C4).682D4, C222(C8.C4), C22.1(C22⋊Q8), C24.4C4.21C2, (C22×C8).166C22, (C23×C4).251C22, (C22×C4).1352C23, (C22×M4(2)).20C2, C2.11(M4(2).C4), C2.11(C23.8Q8), C4.133(C22.D4), (C2×M4(2)).171C22, (C2×C8).34(C2×C4), (C2×C8.C4)⋊7C2, (C2×C4).53(C4⋊C4), C2.14(C2×C8.C4), (C2×C4).1527(C2×D4), (C2×C22⋊C8).34C2, C22.112(C2×C4⋊C4), (C2×C4).748(C4○D4), (C2×C4).551(C22×C4), (C22×C4).274(C2×C4), SmallGroup(128,587)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.10Q8
C1C2C4C2×C4C22×C4C23×C4C22×M4(2) — C24.10Q8
C1C2C2×C4 — C24.10Q8
C1C2×C4C23×C4 — C24.10Q8
C1C2C2C22×C4 — C24.10Q8

Generators and relations for C24.10Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=e2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 236 in 139 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C22 [×11], C8 [×10], C2×C4 [×4], C2×C4 [×4], C2×C4 [×10], C23 [×3], C23 [×5], C2×C8 [×4], C2×C8 [×12], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×4], C24, C22⋊C8 [×2], C22⋊C8 [×2], C22⋊C8 [×2], C8.C4 [×4], C22×C8 [×2], C22×C8, C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×5], C23×C4, C4.C42 [×2], C2×C22⋊C8, C24.4C4, C2×C8.C4 [×2], C22×M4(2), C24.10Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C8.C4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, C2×C8.C4, M4(2).C4, C24.10Q8

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I8A···8P8Q8R8S8T
order1222222224444444448···88888
size1111222241111222244···48888

38 irreducible representations

dim11111112222224
type++++++++--
imageC1C2C2C2C2C2C4D4D4Q8Q8C4○D4C8.C4M4(2).C4
kernelC24.10Q8C4.C42C2×C22⋊C8C24.4C4C2×C8.C4C22×M4(2)C22⋊C8M4(2)C22×C4C22×C4C24C2×C4C22C2
# reps12112184211482

Matrix representation of C24.10Q8 in GL4(𝔽17) generated by

1000
151600
0010
0001
,
16000
01600
0010
00016
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
1000
151600
00015
00150
,
161600
0100
0001
0040
G:=sub<GL(4,GF(17))| [1,15,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,15,0,0,0,16,0,0,0,0,0,15,0,0,15,0],[16,0,0,0,16,1,0,0,0,0,0,4,0,0,1,0] >;

C24.10Q8 in GAP, Magma, Sage, TeX

C_2^4._{10}Q_8
% in TeX

G:=Group("C2^4.10Q8");
// GroupNames label

G:=SmallGroup(128,587);
// by ID

G=gap.SmallGroup(128,587);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,124]);
// Polycyclic

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

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