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G = C23.8M4(2)  order 128 = 27

4th non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

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

Aliases: C23.8M4(2), (C2×D4)⋊3C8, C231(C2×C8), C42(C23⋊C8), C23⋊C818C2, (C23×C4).6C4, C24.44(C2×C4), (C2×C42).15C4, C4.44(C23⋊C4), (C22×D4).18C4, (C22×C4).650D4, C22.9(C22×C8), C22.1(C22⋊C8), (C23×C4).17C22, C22⋊C8.154C22, C23.90(C22⋊C4), C23.160(C22×C4), (C22×C4).423C23, C22.12(C2×M4(2)), C42(C22.M4(2)), C22.M4(2)⋊19C2, C2.2(M4(2).8C22), (C2×C4)⋊1(C2×C8), (C2×C4×D4).1C2, (C2×C4)(C23⋊C8), (C2×C22⋊C8)⋊2C2, C2.7(C2×C22⋊C8), C2.3(C2×C23⋊C4), (C2×C4).1120(C2×D4), (C2×C22⋊C4).17C4, (C22×C4).40(C2×C4), (C2×C4⋊C4).733C22, C22.91(C2×C22⋊C4), (C2×C4).309(C22⋊C4), (C2×C22⋊C4).400C22, (C2×C4)(C22.M4(2)), SmallGroup(128,191)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.8M4(2)
C1C2C22C2×C4C22×C4C23×C4C2×C4×D4 — C23.8M4(2)
C1C2C22 — C23.8M4(2)
C1C2×C4C23×C4 — C23.8M4(2)
C1C2C22C22×C4 — C23.8M4(2)

Generators and relations for C23.8M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, dad-1=eae=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=bcd5 >

Subgroups: 348 in 166 conjugacy classes, 62 normal (26 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×8], C23, C23 [×8], C23 [×6], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C22×C4 [×5], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C24 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×2], C23×C4 [×2], C22×D4, C23⋊C8 [×2], C22.M4(2) [×2], C2×C22⋊C8 [×2], C2×C4×D4, C23.8M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C23⋊C4 [×2], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C2×C23⋊C4, M4(2).8C22, C23.8M4(2)

Smallest permutation representation of C23.8M4(2)
On 32 points
Generators in S32
(1 5)(2 29)(3 7)(4 31)(6 25)(8 27)(9 13)(10 21)(11 15)(12 23)(14 17)(16 19)(18 22)(20 24)(26 30)(28 32)
(2 25)(4 27)(6 29)(8 31)(9 24)(11 18)(13 20)(15 22)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 19)(2 9)(3 14)(4 18)(5 23)(6 13)(7 10)(8 22)(11 27)(12 32)(15 31)(16 28)(17 30)(20 29)(21 26)(24 25)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J4K···4P8A···8P
order12222···22244444···44···48···8
size11112···24411112···24···44···4

44 irreducible representations

dim11111111112244
type+++++++
imageC1C2C2C2C2C4C4C4C4C8D4M4(2)C23⋊C4M4(2).8C22
kernelC23.8M4(2)C23⋊C8C22.M4(2)C2×C22⋊C8C2×C4×D4C2×C42C2×C22⋊C4C23×C4C22×D4C2×D4C22×C4C23C4C2
# reps122212222164422

Matrix representation of C23.8M4(2) in GL6(𝔽17)

100000
010000
0016000
000100
00011160
0011001
,
100000
010000
001000
000100
001011160
0067016
,
100000
010000
0016000
0001600
0000160
0000016
,
4130000
5130000
0067015
001011150
000467
00001011
,
100000
2160000
000100
001000
0000016
0000160

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,11,0,0,0,1,11,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,10,6,0,0,0,1,11,7,0,0,0,0,16,0,0,0,0,0,0,16],[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],[4,5,0,0,0,0,13,13,0,0,0,0,0,0,6,10,0,0,0,0,7,11,4,0,0,0,0,15,6,10,0,0,15,0,7,11],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0] >;

C23.8M4(2) in GAP, Magma, Sage, TeX

C_2^3._8M_4(2)
% in TeX

G:=Group("C2^3.8M4(2)");
// GroupNames label

G:=SmallGroup(128,191);
// by ID

G=gap.SmallGroup(128,191);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,851,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*c*d^5>;
// generators/relations

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