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G = C23.30D8order 128 = 27

1st non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.30D8, C24.146D4, C23.15Q16, C23.28SD16, C22⋊C86C4, (C2×C4).6C42, C22.32C4≀C2, (C22×C4).22Q8, C23.42(C4⋊C4), C2.C429C4, (C22×C4).638D4, C22.1(C4.Q8), C2.8(C426C4), C22.1(C2.D8), C23.7Q8.2C2, C22.34(C23⋊C4), C2.6(C22.4Q16), (C23×C4).192C22, C2.3(C22.SD16), C2.6(C23.9D4), C22.38(D4⋊C4), C23.214(C22⋊C4), C22.28(Q8⋊C4), C2.3(C23.31D4), C22.44(C2.C42), (C2×C4⋊C4)⋊1C4, (C2×C4).70(C4⋊C4), (C2×C22⋊C8).5C2, (C22×C4).149(C2×C4), (C2×C4).299(C22⋊C4), (C2×C2.C42).11C2, SmallGroup(128,26)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.30D8
C1C2C22C23C24C23×C4C2×C2.C42 — C23.30D8
C1C22C2×C4 — C23.30D8
C1C23C23×C4 — C23.30D8
C1C2C22C23×C4 — C23.30D8

Generators and relations for C23.30D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=a, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=abd-1 >

Subgroups: 320 in 138 conjugacy classes, 48 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×7], C22 [×4], C22 [×12], C8 [×2], C2×C4 [×4], C2×C4 [×30], C23 [×7], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C22×C4 [×6], C22×C4 [×14], C24, C2.C42 [×2], C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4, C2×C4⋊C4 [×2], C22×C8, C23×C4, C23×C4, C2×C2.C42, C23.7Q8, C2×C22⋊C8, C23.30D8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, C23⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C4.Q8, C2.D8, C22.SD16 [×2], C23.31D4 [×2], C426C4, C22.4Q16, C23.9D4, C23.30D8

Smallest permutation representation of C23.30D8
On 32 points
Generators in S32
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(2 10)(4 12)(6 14)(8 16)(17 29)(19 31)(21 25)(23 27)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(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 8 18 17)(2 28 19 15)(3 14 20 27)(4 22 21 5)(6 32 23 11)(7 10 24 31)(9 16 30 29)(12 26 25 13)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim111111122222224
type+++++-++-+
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16C4≀C2C23⋊C4
kernelC23.30D8C2×C2.C42C23.7Q8C2×C22⋊C8C2.C42C22⋊C8C2×C4⋊C4C22×C4C22×C4C24C23C23C23C22C22
# reps111144421124282

Matrix representation of C23.30D8 in GL6(𝔽17)

100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
3140000
330000
0051200
005500
0000016
000040
,
3140000
14140000
0012500
005500
000001
0000160

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

C23.30D8 in GAP, Magma, Sage, TeX

C_2^3._{30}D_8
% in TeX

G:=Group("C2^3.30D8");
// GroupNames label

G:=SmallGroup(128,26);
// by ID

G=gap.SmallGroup(128,26);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,3924]);
// Polycyclic

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

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